When Shakespeare uses "Clown" as a character name in a play, he is noting the fact that this character doesn't need a name and that the part is to played for comic effect. In so doing, he was calling for the comedian of the troupe, the actor with the great timing and delivery for humorous kinds of lines.
Your essay will depend a lot on which play you are addressing. In most plays where there are clowns, the clowns serve as comic relief and sometimes a bit of common sense intelligence in light of all of the tension that surrounds them. Two of the most famous clowns are the two gravediggers in Act 5 of Hamlet. The two grave diggers are completing Ophelia's grave in preparation for her funeral later in the day. In the course of their conversation and joking, they reveal to the audience that Ophelia is only receiving a proper Christian burial because the King decreed it. So this clowning serves to address that issue and advance the plot. Once Hamlet appears, unknowing of Ophelia's death, he and the gravedigger have a funny but poignant conversation about what happens to people after they die. In the course of the conversation, Hamlet realizes that no matter who a person was in life--King, Julius Caesar, Alexander the Great--when he dies, he just becomes dust again.
This realization that death is a great equalizer actually frees Hamlet from some of his trepidation over taking action to avenge his father's murder. He has talked about death and dying the whole play but here, talking to a common, comic man, he finally understands death and dying. The purpose of the clowning conversation is to develop a new side of Hamlet so the audience is ready for the end of the play. We have had a clowning break and are ready for what we know will be a bad situation.
There are Clown (rural and accidentally foolish) and/or Fool (urban and intentionally foolish) characters in several other of Shakespeare's plays. In order to write about them, you need to brainstorm what thematic function they serve and how they are used within the plot line of the play.