The Waiter Gremlin was one of the many gremlins that invaded the Clamp Center building. He is seen serving the rest of the gremlins as a waiter and wearing a black tie, a towel around his waist and towels on his arms. It is unknown why he's doing that because Gremlins are evil and would not like to do catering service, but then, it could be for a reason.
Near the end of the film, he is seen standing behind the bar counter in the lobby of the Clamp Center and plucks a bellpepper from the Vegetable Gremlin to put into a martini for another gremlin and he gets squirted with water by that gremlin. Later on, he is destroyed along with most of the other gremlins (except Greta) when they were electrocuted by the Electric Gremlin.
Although its hard to notice, there is a very similar gremlin in Dorry's Tavern in the first film. He is seen walking around and serving different gremlins, much like Kate herself.