An Amusing Alien is a sci-fi, fantasy, or otherworldly characternote "alien" is not meant to be exclusively the modern extraterrestrial meaning whose main purpose is to amuse the audience.
While the Amusing Alien might have a serious (or mundane) task in the story, a character belongs in this trope only if comedy is his primary reason for existing in the work. An elf archer who tells puns incessantly is not an Amusing Alien, whereas an elf archer whose arrows always take an unexpected path to hit their targets would be. If the character becomes irrelevant, uninteresting, or The Load when played seriously, then he's an Amusing Alien.
Most of the time, the Amusing Alien will be a secondary or background character, to avoid excessively derailing the plot with their antics. The primary exception is in comedies, where casting an Amusing Alien as the main character leads to lots of easy jokes.
May overlap with Funny Foreigner, but the two tropes are not identical. An otherworldly Funny Foreigner elicits laughs from their ignorance and bizarre behavior; an Amusing Alien, on the other hand, can be anything from The Smart Guy, a Deadpan Snarker, or a Genre SavvyFourth Wall Observer. As a general rule, if the Amusing Alien is of a race or species that's common in the work, the trope is Funny Foreigner instead.
If there's a multi-character party, the Amusing Alien will probably be the Team Pet, the Non-Human Sidekick, or the Token Non-Human.
Given the subjective nature of comedy, the risk is high that an Amusing Alien can end up becoming a Scrappy if not handled carefully. This trope documents alien characters that were intended to be funny; whether or not they succeed is another matter...
Also see Alien Among Us, Pet Monstrosity, Plucky Comic Relief, Fantastic Anthropologist, and Crazy Cultural Comparison. Supertrope to Funny Robot.
The monsters of Fin Fang Four (Gorgilla, Fin Fang Foom, Elektro and Googam) probably qualify: they're former monsters who are "rehabilitated" and do mundane jobs (window washer, parking lot attendant) at the Baxter Building.
An hilariously out-of-the-ordinary example of this trope occurs in Diane Duane's Young Wizards novel Wizard's Holiday, when Dairine attempts to lead her three alien wizard guests on a trip to the mall. Considering that one of them is a tree, one a giant centipede, and the third a sun prince who looks like an anime character, you can guess what kinds of Amusing Alien antics will happen.
Animorphs the Iskoort are a species of amusing aliens. (and annoying too)
Sometimes Ax falls into this trope, though he isn't exclusively it.
The Pakleds start off as a straight example of this trope but actually use their ignorant appearance to deceive others.
Double subverted by the Pakleds, actually: they're bright enough to steal technology and kidnap Geordi LaForge but not bright enough to actually use their stolen technology without the help of a kidnapped engineer...nor to see through the technobabble and bad acting the Enterprise crew uses to effect a rescue.
And they are slow - just not quite as much as people tend to think
Morn, the perpetual barfly at Quark's Bar on Star Trek: Deep Space Nine, is a meta example; he's described as an extremely talkative being who knows the funniest joke in the universe, and others have attributed various fantastic feats to him. Even so, he's always shown in stoicsilence on the show.
Meego, the Shape Shifter alien from the CBS sitcom of the same name.
Dex, also known as the Masked Rider, has traces of this at times. In one example would be that his race, Eltarians, derive from insects instead of apes. Now guess what he answered on an Earth biology test on the subject of human origins…
Tripitaka's dragon-cum-horse-cum-human from Monkey.
Everybody in Hidden Hills on The Neighbors, though whether the Zabvrons or the Weavers are the "aliens" depends on how you look at it.
Doctor Who has recently acquired one in the shape of the Sontaran bonehead Strax. He has difficulty telling the difference between human genders and once declared war on the moon.