Cute Monster

He's like a big puppy. From from another world. That spits corrosive acid.

Some monsters are simply animals. They might be unusually smart or have extra limbs or even heads, they might be bigger than their real-world counterparts, but in the end they're clearly the domain of zoologists and concerned mainly with food, territory and mating. Some monsters go out of the way to make life difficult for the hero or humans in general, they are Exclusively Evil. Some monsters are so powerful and dangerous they are beyond the very comprehension of mortal men; the wisest course of action on encountering one of those is to run, and even then the odds are against you.

And then there is another kind of monster. Maybe they're just a baby, and someday they'll grow up into one of the first types of monsters, but until then they're not so bad. Maybe they've discovered The Power of Friendship. Maybe they try to be evil, but they're not very good at it. Maybe in-universe they're horrifying, but the audience gets to see the softer side of them. But whatever the reason, these monsters aren't very monstrous. Simply put, they're cute.

Related to Cute Monster Girl, but where those are generally cute in a sexualized way, these are generally cute in the sense of kittens, puppies and babies. Fanfic focused on Cute Monsters is generally less focused on relationships between them and humans than fanfic focused on Cute Monster Girls; sure, it happens sometimes, but not as often and when it does it's less supported by the Canon. As mentioned above, some of these also count as Ridiculously Cute Critters, but these are all still monsters, and they just happen to not be dangerous, or not dangerous to the right people, or only in an endearing way.

Examples of Cute Monster include:

Comic Books

X-Men: Lockheed the dragon. Most of the time. He's a purple dragon, but the size of a dog or even just a housecat. He's treated as a pet by Kitty Pryde, and he's fine with that.

Film

The Wizard of Oz has the Cowardly Lion. He's a monster, attacking Dorothy and friends in the woods, but as soon as he's startled by a dog barking he needs cheering up and help with his own problem as much as Dorothy's party needs help with theirs.
In Men in Black the newborn alien that Jay helps deliver. Presumably mom's tentacles beat the everloving crap out of Jay during labor, but the resultant infant is downright adorable. And a squid.

Literature

The dragons of the Temeraire series. They're roughly as intelligent as humans and most of them grow to be massive, but they're generally flighty and very compassionate to their human companions. And while they're intelligent, they only have the wisdom and common sense appropriate for their life experience, and most dragons we see are just a few months or years old. Thus there are many scenes where Temeraire has to be told politely but firmly that he can't just eat any cow he sees, they should be bought and paid for, even if it seems unfair to him that he can freely eat any fish he catches; and despite his size he makes slower progress than a human while walking because he's curious about every unfamiliar rock, tree and house he sees; and they think of human babies as eggs, and it's painfully awkward to explain to them that humans give birth differently; and feral dragons jockey for status among themselves on the most puffed-up pretenses.

Multiple

Mons.

Tabletop Games

Pseudodragons in Dungeons & Dragons. Miniature dragons the size of housecats and with similar personalities, although they have human-level intelligence and the draconic love of shiny things.

Video Games

Kog'Maw, as pictured above. He's an alien omnivore from the same dimension that spawned Cho'Gath and drove Malzahar to insanity. Despite this Kog'Maw isn't as much evil as he is hungry, and the combination of his innocent mannerism and endearing behaviour(which includes chasing his own tail) has won the hearts of many players. Not to mention the fact that he is one of the strongest characters in the game does help.

Web Comics

Cerberus in Zeus and Sons.

Western Animation

A number of characters in Shrek, but especially Dragon and the dragon-donkey hybrids even more so. At the end of the second movie and through the third they are small, round fuzzballs, barely flying on tiny wings, randomly breathing tiny, almost-harmless jets of flame.
The vegetarian sharks in Finding Nemo. They have self-control problems when they smell blood, but the rest of the time they are all too eager to help the protagonists with their problems.
Nibbler. An alien about as big as a dog, with three eyes and a distending jaw and massive appetite who poops starship fuel, kept as a pet by Leela. Later in the series revealed to be a member of a highly intelligent alien race, but still a big eater and happy to be kept as a pet as long as his litter box is cleaned regularly.
The titular monster, Francœur, in A Monster in Paris.
Insectosaurus from Monsters vs. Aliens. A fuzzy baby caterpillar the size of a city block.
Almost all the monsters in Monsters Inc.. That's the point.