Today we have yet another linguistic bone to pick with people who can't seem to get their English grammar right. While you may pronounce the words elicit and illicit the same way, they do have different spellings which you should learn to use correctly, since they mean completely different things!
This verb comes from the Latin term elicitus, and means to evoke, inspire, or draw something out of someone, whether a reaction or information. If you're crying because of relationship troubles, you may elicit someone's sympathy. We can only imagine that Lady Gaga's crazy outfits are designed to elicit reactions from us. You can also try your hand at eliciting laughter from someone by telling them a joke.
Note the initial i and the double l in this adjective's spelling! Things that are illicit are illegal, not permitted, or illegitimate. If a noun has illicit in front of it, then you should know to stay away from it. Illicit affairs are never a good idea, whether you take the phrase in the romantic or the business sense. It's also a bad idea to get involved with illicit drugs, as well as downloading illicit copies of films or music.
In summary, you may want to elicit certain behaviors, but not if they're illicit! Unless you want to end up in prison, that is.