Wizard Classic

While the term wizard itself is often applied as a catch-all word for male magic users in fiction, the image of the classic mage as it has evolved out of Western traditions and folklore has a number of distinct trappings and tropes. In short, this is the character most people think of when they hear the word "Wizard":

Traditional Wizardly Attire: Wizards are known for wearing long robes, especially in blue or grey, and pointed hats, often floppy and without brims. Sometimes the hat is omitted, but rarely the robe. They also have a long Wizard Beard that's usually white.
Carry a Magic Staff: Wizards are known for carrying staffs which they use as a device to aid in casting spells and for a walking stick. They are heavily associated with wizards over any other type of magic user.
Wizardry: Wizards practice wizardry specifically as their type of magic which tends to have to do with cosmic energy and powers through intensive study of arcane knowledge. They are often seen reading the stars and tapping into the energy of the earth directly, casting spells with runes or magic words, or pure thought. Wizards appear to have been somewhat based on Druids and ascetic Hermits but are not directly comparable. They may have a personal Spell Book, but more likely they have an entire library of gathered knowledge. The most powerful wizards may call themselves Archmage.
Wizardly Habits: Wizards are very often solitary folk who live alone in far off, difficult to reach places, studying in seclusion. They also tend to live in tall towers, the better to be closer to the stars. They have been known to enjoy Walking the Earth, however, and people seem to stumble upon them at random...or is it fate? They also tend to have apprentices hanging around. A Wizard can be a Court Mage but even then they tend to have their own areas and act aloof. If they have a pet, it's usually a bird of some kind, like a crow or owl. A degree of eccentricity is quite common, be it real or feigned, which masks the wizard's considerable power.
Mostly Male; Wizards, when the title is not just a term for 'male magic user', are still often shown as being Always Male. This may have more to do with males being the more educated of the sexes in historical times when even wealthy and noble females were rarely even taught to read or that women can't grow beards. For these reasons alone the possibility of a female wizard would be unlikely. However, if wizards can be either gender, and are different from a witch, it is common for the female variety to be called a Sorceress or have some other term applied to help with the differentiation. Strangely Wizardess or some other variant of the word "wizard" is practically non-existent as a term.
Association with Age and Wisdom: Wizards are usually very old, the implication being that it takes one that amount of time studying before one can really be considered a true wizard, fit to master the secrets of the universe. Sometimes these wizards cast illusions to appear old, if they aren't, so they're more respected. Furthermore it's common that Wizards Live Longer than most other people, even most other magic uses, for which this is common. Wizards are known for being very wise, and 'wizard' actually comes from the word 'wise'. They often act as advisers to very important people, like royalty, and mentors to heroes.

Wizards can be of good or evil alignment, but when they do turn evil, it's common for them to have a different term to denote this, such as Sorcerer, though other times the two are interchangeable or denote different types of magic-users. It's common for a Classic Wizard to be contrasted with a Classic Witch, in works where they are significantly different types of magic users and not gendered titles. Wizards seem to be a bit less popular outside western fantasy media than the witch, probably because they are a bit harder to make cute, or hot.

A Classic Wizard is not always a Squishy Wizard, but it's common. The same with a Kung Fu Wizard. See also The Archmage.

Examples of Wizard Classic include:

Literature

Gandalf and Saruman in The Lord of the Rings. Both magic, bearded old men who were long-lived and nigh immortal and carried staffs, and gave advice. Gandalf walked the earth but Saruman lived in a tower.
Dumbledore in Harry Potter. While all male magic users are called 'wizards' in the series, Dumbledore specifically fits the 'wizard' image. He's old, wise, has a beard, wears robes and floppy hats, and kind of lives in a tower too. Uses wands, although so does everyone else in the universe.
According to the Word of God, Harry Dresden from The Dresden Files is half-this (half-Gandalf, to be exact), half-Sherlock Holmes.
The Wizards of Unseen University in Discworld fall into this, as does Rincewind (whose association with the university varies.) Of course their "wisdom" in regards to magic mostly consists of not using it and jockying for a Klingon Promotion or two. With the exception of a few who seem to actually do important research and get results (for example creating Hex and Roundworld).
The 500 Kingdoms, being a gentle subversion of fairy tales, has both Wizards and Sorcerers that fit this mold, although Sorcerers tend to do the lofty tower isolation bit and Wizards tend to do the wandering the earth bit. Note that neither title denotes evil, though - that just gets you an Evil or Dark tacked on the front. Justified in-story by the fact that magic is an semi-active force that likes things to fall into Traditional paths, so the more like a Traditional wizard you look and act, the better.
In The Dark Lord of Derkholm, Wizards are pretty diverse folk and include both genders, but Mr. Chesney made all wizard-guides have to conform to look as close to the classic version as possible when they headed tours, including making all wizards grow beards.
In the Enchanted Forest Chronicles, all wizards wear traditional wizardly attire, have beards and carry staffs, which suck up magic from their surroundings. All wizards are pretty much bad guys in this incarnation.
- Supposedly there are good wizards (who aren't part of the organization causing all the trouble throughout the series), but they're only mentioned in passing at the end of the fourth book.
Disciples of Aldur in the Belgariad.
The Dorrie the Witch stories have featured a few, some good and some evil.

Mythology

Depictions of Merlin in almost all media.
Also Odin from Norse Mythology, a god of wisdom who was the precursor to Merlin and carried a spear instead of a staff. Stories of him wandering the lands disguised as a simple traveller were a direct inspiration for Gandalf. He thus serves as the Ur Example to many later characters in this tradition.
- In terms of wardrobe, however, Odin is more famous for a cloak and a wide-brimmed hat.

Tabletop RPG

Magic users (wizards) in early editions of Dungeons & Dragons.
- Forgotten Realms has Elminster traditionally portrayed this way. It's demonstrated in novels (and confirmed) that he's a man of many masks and looks like this mostly in the "example for younger wizards" role because he knows lots of folk look up to him. He's not inclined to play by rules unless it's the whole point, and isn't always recognizable as a living creature, let alone himself, when goes out to do something.
  - Marune of the Shadow Thieves (see in the canon art).

Video Games

Of all the mage characters in Dragon Age Origins, First Enchanter Irving probably fits this image best.

Webcomics

Homestuck: Zazzerpan from the in-story novel The Complacency of the Learned. Other wizards are mentioned as well, but there's not enough information to know how closely they stick to the archetype. The Lalonde house is decorated with a frightening amount of artwork of Zazzerpan and numerous other classic wizards—these are the cause of a bizarre, passive-aggressive (and completely one-sided) fight between Rose and her mom.