When you elicit, you're bringing out a response of some sort. A good comedian elicits a lot of laughs.
Elicit has to do with creating or provoking a response. A great speech will elicit cheers — a bad speech will elicit boos. Teachers try to elicit responses from students. If a friend smiles at you, it will probably elicit a smile of your own. In court, a lawyer might try to elicit mistakes and inconsistencies in the testimony of a witness. In all cases, whatever is elicited is some kind of response.