I think one of the things that makes Shakespeare's comedies unique is that they have some pretty serious concepts along with the comedy. None of them are pure slapstick, all jokes and goofiness and no substance. There is some meat on the bones.
The above is all true; however there were specific elements of language that Shakespeare used to create comedy in his work, such as the use of the pun, i.e. the play on words and other forms of figurative language, such as euphamisms. His use of language is what makes his comedies and oeuvre in general, unique. I would argue that in addition to the implementation of comic relief characters within his romances, histories, and tragedies, his use of language and ablity to manipulate words, to have fun with the English language, is where the real humor emerges from.
Actually, they are not! Other writers for "The King's Men" and other theatrical troupes were writing other plays, and Shakespeare's comedies were not really "unique" (one of a kind). For example, one of his primary rivals was Christopher Marlowe, famous for his "Doctor Faustus" - a humorous play in which in exchange for riches and fame, a man sells his soul to the Devil.
One reason, though, that Shakespare's plays were so popular was that he wrote in a script that had something in it for everybody. Some of his humour and more serious passages are subtle and erudite; other parts are quite bawdy, often written in the dialect of the lower classes. In this respect, Shakespeare had real versatility and was a marketing ace.
Otherwise, a lot of aspects are "borrowed." Some of his more famous works such as "Hamlet" and "Romeo and Juliet" were based on stories already circulating. His histories were evidently inspired from great events in the past. Shakespeare also used a lot of elements from Greek drama set forth as criteria for a "good play" (particularly true of his tragedies: example, the idea of tragic flaw and the use of a chorus).
Perhaps the most distinguishing mark of Shakespeare was his ability to weave humour intermittantly into even in his most serious works. As heightened suspense cannot be maintained indefinitely, he used comic relief to structure the story line into rising and falling action for optimal effect.