Elicit is a verb. It means to evoke or to draw out.
Listening to music by Stan Rogers elicits fond memories of my life in my early twenties.
A teacher often asks questions to elicit answers from his/her students.
Illicit is an adjective. It means illegal or something that is disapproved of for moral reasons.
Cocaine is an illicit drug.
Some christian groups believe that Marilyn Manson's lyrics are illicit, and have tried to ban the group from playing in towns all across the United States.
Think you won't remember which is which? Here's how I remember:
E = Evoke = Elicit
I= Illegal = Illicit