They are capable of changing human beings into other vampires. Classical vampires like Dracula needed to go through a more elaborate process to make another vampire, but bowdlerized versions removed the detail where he made the victims drink his blood to begin the transformation, leading to the idea that victims can become vampires from just a bite.
Modern versions that don't have such a process often blur the line between vampire and zombie, sometimes leading to a full-on Vampire Apocalypse because of a runaway Viral Transformation. Worse, sometimes Vampires who don't keep fed turn into Zombies.
Recently, the idea has arisen that vampires judge each other by how far removed they are from a "source". The highest social status belongs to someone who somehow became a vampire without being turned by one via bite.
Of course, there can also be a fusion of "types". A vampire may create mindless undead slaves via simple feeding; (often referred to as "spawn") but to create a thinking vampire with the potential for the gambits of powers, the full process is needed.
Or they create living servants like ghouls or blood-slaves who feed on their blood, get power from it some how, and protect their masters any way they can.
Vampires are almost always inhumanly strong, fast, and durable, often to the point of being Immune to Bullets and most other mundane weapons. For some, especially more modern ones, this is where it ends, making them effectively little more than intelligent (and stylish) super-zombies.
A variation of this is to give them their own unique "gifts" (telepathy, for example) that make them more distinct from their brethren, though all share the same aforementioned set of "normal" vampire powers.
The original folklorish vamps were either disease ridden monstrosities or soul-sucking ghosts; in either case, their mere presence was likely to harm you, and though you could ward them off at night you couldn't actually kill them until the daylight hours, and sometimes you couldn't properly kill them at all since, being evil spirits, the best you could do is stop them from coming back.
The traditional Victorian vampire has a range of supernatural abilities. Dracula had shape-shifting, limited flight, control over animals and the weather, the ability to scale walls and other gifts, on top of the standard vampire strengths. It is unclear if this is due to Dracula studying Black Magic to enhance his skills (and this type of vampirism can come with an innate ability to learn that as well- its also implied that this may have been how Dracula became a vampire) or if it was due to his advanced age. Its possible that both might be true.
The strength of a vampire can sometimes be determined by its age, with older vampires usually (though not always) being stronger than younger ones. Sometimes this merely means that they are stronger and harder to kill, if it means anything at all. Some may evolve (or de-volve) into something closer resembling some progenitor vampire race, which can occur either gradually or in spurts, which makes them yet more superhuman.
In other cases the vamp can age into an outright Humanoid Abomination which will usually mean they are much more powerful, though some may understandably lament their transformation into outright monstrosities and more obvious loss of humanity; this, again, may happen gradually or in spurts. The ones who won the Superpower Lottery have, either naturally or through using their immortal lifespan to acquire ridiculous amounts of magical power, evolved into outright Gods of Evil, and are a menace to the entire world.
Wooden stake through the heart. In most modern depictions, this is fatal; in the original folklore, it merely stops the vampire from leaving his coffin. In most of the older stories, one had to use a hammer or a grave digger's shovel to drive the stake in, which meant that vampire stakings mainly happened during the day when the vampire was asleep, but recently, it's become oddly easy to do by hand. Remember, the ribs are there to prevent just such an occurrence. In some cases a special specimen of wood is needed for the stake to be effective, commonly Hawthorn, and occasionally it needs to be blessed or enchanted, but not all vampires are this picky about what goes through their chests.
Decapitation - and as a bonus point, this one isn't exactly exclusive to vampires; it conveniently works on almost any supernatural creature, and humans too. Then again, so does a stake through the heart...
Fire - another one that can be used to deal with most other supernaturals and also humans, although it varies between interpretations on just how much you need. Really, the only common Achilles's Heel definitely unique to vampires is...
Direct sunlight. Originally, they actually had to sleep in their coffin during the day, and sunlight wasn't fatal. They were merely dormant during the day, making it "easy" to sneak up on them. Nowadays, they just hole up inside, and sunlight literally has the power to make them spontaneously combustive. Sometimes this is specifically ultraviolet radiation; sunlight is dangerous, but a light-bulb is not. This can vary with age, either becoming less or more effective over time. Because of its lethality, some vampires choose Suicide by Sunlight.
Dracula in Bram Stoker's novel was almost unaffected by sunlight; it limited his Shapeshifting powers but he could still walk around, was still super-humanly strong, and definitely wasn't burned. The same goes for other vampires before Stoker's, such as Carmilla and Varney the Vampire.
An interesting inversion are Arabian vampires. They're active during the day and sleep at night, since people were naturally more afraid of the daytime in the desert.
The idea that sunlight isn't fatal has undergone somewhat of a resurgence. The vampires in L.J Smith's Night World series can survive exposure to sunlight, but it inhibits their powers. The vampires in Moonlight can survive exposure to sunlight for a limited amount of time. In some folklore, vampires were actually at their strongest at high noon, when their shadow was at its smallest. They were weakest at dusk, when their shadow was at its longest.
If they exist in the story, magical weapons or other supernatural creatures might also have special abilities to kill vampires.
Attempting to cross flowing water (e.g., rivers and oceans). Frequently interpreted to mean vampires can't cross flowing water. The effects of flowing water vary greatly depending on the story. Dracula, for example, could cross running water at the slack or flood of the tide.
Crosses, but not necessarily other religious symbols. Originally, it had to be a full blown crucifix (that is, a cross with a figure of Jesus on it). In modern renditions, this is usually subject to the power of belief of the wielder, the vampire, both, or neither. For instance, if a character is a sincere Jew, then they could use the Star of David to ward off a vampire. Then you can have a vampire who carries his own crucifix, as he is a believer too, like Henry Fitzroy in Blood Ties. He also prays and goes to confession (he figures that he is subject to the same sins as humans, and needs to do penance for them). Fortunately, he is a Friendly Neighborhood Vampire. In other cases, the religion the symbol represents has to have been around during the vampire's lifetime to have any effect. If a vampire predates all modern religions, don't go reaching for your crucifix.
Holy water (drinking or total immersion in holy water often IS lethal)
Communion Host (in Bram Stoker's novel, it was used to seal a crypt and prevent a vampire from entering its coffin at sunrise.)
White roses / roses in general (might have connection with beliefs that roses will not grow over grave)
Garlic or Onions, although this was more to ward off vampires, not harm them.
Thorns (especially hawthorn) in Middle- and Eastern European folklore
Wolf's Bane (Aconite), a plant featured prominently in the 1930s Dracula film, but also Foxglove (Digitalis) and Holly Bushes.
Also, folklore tells us vampires get disoriented (or even driven mad) At the Crossroads, and cannot tell one direction for another. Urban vampires seem to have developed a strong resistance to this weakness, especially those that frequent downtown districts (probably by building up an immunity from all the intersections).
In some folk lore, vampires are all stricken with a debilitating obsession with numbers, if you throw a quantity of small objects on the ground in front of them (seeds,grain,beads etc.) they will not be able to resist the urge to pick it up and count it, this affords the victim time to either run away or kill the vampire. ("Three! Three mustard seeds! Muha-ha-ha!"). Putting said objects into a vampire's coffin keeps them busy counting as well. Sometimes, the urge is powerful enough that you can force the vampire to expose itself to dawn. Sometimes it is not an "urge" but they are somehow forced to count those objects.
No reflection (often because the vampire has no soul, but see below). This sometimes extends to shadows. But it depends on the vampire apparently. In one medium there are several types of vamps who have various weaknesses. In more recent examples this has been 'modernised' in terms of the vampire not being able to be picked up by audio or video recording or transmitting equipment.
No heartbeat/breath. Sometimes the no breath thing means they can't do mouth-to-mouth resuscitation, but if they can talk, they must be able to take air into their lungs and expel it intentionally. Little logic problem there, but we're talking about vampires so logic should take the day off. However Vampires do not need to breathe, it seems to be a reflex they have left over from being alive hence they can never drown, suffocate, or be poisoned.
Physical features, such as being exceedinglypale, having unusual eyes (Glowing Eyes of Doom/Hellish Pupils/Animal Eyes), and, of course, Fangs Are Evil. In folklore, there were numerous physical telltales - eyebrows that met over the nose, fingers all the same length, hair in the center of the palms or backward-facing palms - that are mostly overlooked in modern versions. The original novel-version Dracula has practically all of them. If they can hide some or all of them, dropping the disguise constitutes using Game Face. Sometimes vampires will become more and more human-like in appearance as they consume more blood/live longer. Sometimes... not.
Body temperature: Vampires, being dead, are almost always at room temperature or colder.
Technically, they are dead. Pretty spry for a dead guy, though.
Vampires don't age as we mortals do. Sometimes, this is genuine eternal youth. Sometimes long periods of time undead can result in a pretty inhuman-looking character. Sometimes, they age like us, just at a much slower rate.
Related, they usually suffer from Creative Sterility in both the biological and artistic sense. They can not beget any children unless it's a male vampire and a live woman, in which case a Dhampyr is the result. They may however be capable of turning a child into a vampire, which results in an ageless Undead Child. If it's a "living" vampire species this is usually waived.
Rarely, the vampire is immortal but must restore his/her youth by drinking blood. In abstinence, they "age", and immediately begin to grow young after they've fed. This originated with Dracula and with persistent stories about one Elizabeth Bathory's bathing habits.
A work will usually address these baseline rules even if they're not enforced. Sometimes an unused rule will be explained away as a Fake Weakness propagated by the vampires themselves.
Some folklore claim the only way to permanently kill a vampire is to hammer a stake through its heart, shove garlic in its mouth, cut off its head, tear off its ears, dismember it, burn the pieces in a fire, and then scatter the ashes across holy ground. This will also permanently kill most anything, including pale spooky goths who happen not to be vampires. A few old folklore suggest that even this only works until a full moon shines on the ash. All this on the theory that vampires were corpses animated by evil spirits. Doing all these things rendered the corpse unusable by the spirit. By contrast, the easiest supposed way to stop a vampire is finding his coffin and turning him face down to make him "bite the dust, not people".
Cannot be photographed or caught on video, often an extension of the "no reflection" rule. This may also be related to the silver rule; mirrors and photographic film are both (usually) made from silver.
In Moonlight, Mick explains in a voiceover that he could not be photographed when silver was used in film, but digital cameras have changed all that.
In the TV series Ultraviolet (unrelated to the film), the vampire hunters use sights that pretty much amount to video cameras strapped to their guns in order to tell vampire from non-vampire.
In the anime Magical Pokaan, Pachira does not show up on a normal digital camera but is perfectly visible when viewed with an infrared camera.
Cannot be heard over phone lines.
If there are any actual Holy Relics, these things will kill a vampire even if they're just in close proximity. However, these are rarely used. Some variations have the relics only being effective when the faith of the wielder is strong. In other variations, the relic is only effective if the vampire believes that it can harm them.
Can turn into bats, wolves, or wisps of smoke for travel. (Bats are by far the most common.) A rare transformation featuring prominently in early literature (such as Dracula) was the ability to turn into elemental dust in moonlight.
A connection to bats isn't part of older vampire folklore because all vampire bats are native only to the New World.
Can turn into other creatures that drink blood: vampire bats, mosquitoes, ticks. (Sometimes they become a single creature, more rarely a whole flock/swarm.)
If killed, can be restored to unlife with the proper procedure. One early version of this, appearing in both pre-Dracula stories The Vampyre and Varney the Vampyre, is that a vampire will be revived and healed automatically if its corpse is bathed in moonlight.
Must sleep in the soil from their homeland/original grave.
There are two social profiles for vampires. The first is a loner who may keep a cadre of vampire slaves and possibly a mate. Dracula fits this profile. The second is a "vampire society" where houses of vampiric lineages act and compete within a Masquerade.
Level of "deadness" varies. On one side of the spectrum, it's just lack of heartbeat and skin that's cool to the touch. On the other, they're literally a moving, rotten animated corpse.
Occasionally suffer from severe OCD. One folkloric method of dealing with Vampires was to drop thousands of grains of rice in their coffin, the theory being they'd be compelled to count them all when they awake, wasting the whole night instead of getting up and terrorizing people.
The folklore version also is told with sesame seeds, and may also extend to any small, numerous nut or grain, if not any particulate (handfuls of sawdust?). Fairies also have this problem.
Dropping a bunch where you stand is a known way to escape the OCD variant of vampire.
A similar folklore variant involves hanging a sieve, colander, or other household item that's full of holes outside your front door. That way, the vampire will stop and count all the holes, leaving them vulnerable at sunrise. ONE! TWO! THREE! FOUR! FOUR GRAINS OF SAND! AH-HA-HA-HA!
Apparently poppy seeds were used to great effect in Greece, as they had the additional benefit of putting the vampire to "sleep".
Also on the OCD theme, vampires will, like fairies, be obsessed with out of place and messily-tied knots, and must stop what they're doing to untie them.
May or may not beat war with werewolves. If there are werewolves (or other supernatural beings such as The Fair Folk) around, attempting to mix the two (by 'converting' a werewolf into a vampire) may be impossible, dangerous, or simply against the rules of The Masquerade. In the case of Faeries, Demons or similar otherworldly beings, drinking their blood will generally cause the Mushroom Samba, possibly combined with strange random supernatural effects such as precognitive flashes or a delirious walk in daylight with no other ill effects. This differs in folklore, where vampires often have the ability to turn into wolfmen, and werewolves who are killed can return as vampires.
Sometimes instantly turn to dust or dissipate completely when killed, an idea believed to have first turned up in Stoker's Dracula. This may ignore mass-energy conservation, as in Buffy the Vampire Slayer, or release enough energy to cause serious damage to anything nearby, as in Ultraviolet and From Dusk till Dawn.
The purpose of vampires in the story varies quite widely. They serve as the Big Bad or as a metaphor for something (communicable diseases like plagues or STDs; alcoholism, drug addiction, denial of aging). There is some danger of the vampire character being too on-the-nose for the metaphor.
The "baseline rules" above are strongly influenced by Hollywood tradition, and not "real" vampire folklore, or even classic vampire fiction. For instance, as (properly) shown in the 1992 Dracula with Gary Oldman and Winona Ryder, and in 2003's League of Extraordinary Gentlemen, Dracula and other "folkloric" vampires were at the most inconvenienced by sunlight, not killed instantly.
In Stoker's novel and earlier vampire lore, sunlight did not cause vampires to go up like flash paper. Several times in the novel, Dracula appears in broad daylight with no ill effects. He is simply incapable of using at least some of his vampiric powers during the daylight (he cannot change form except at dawn, noon and dusk, but still seems to be able to charm wolves to some degree). Sunlight causing a vampire to suffer pain and damage, glitter, smolder, or go up like a one man pyrotechnic band was pretty much wholly created by Hollywood, and specifically, by F.W. Murnau in Nosferatu, the first film to use this idea and probably its inventor (which incidentally was made in 1922 in Germany, when a typical Hollywood film consisted of Buster Keaton being chased by a bunch of coppers).