'You are welcome' or "you're welcome" is the response to "Thank you" or "thanks". Cheers is like 'see you later', 'goodbye', or 'bye'. So you can say cheers in response, or one of the other examples.

To clarify one thing from the answers above: In the U.K. and some other countries, "Cheers" is commonly (informally) used to say "thank you" That usage is not part of American English. I'm American, but I'm pretty sure that "You're welcome" is a fine response if someone thanks you using "Cheers."

Further to Cory's answer, "Cheers" is often used when 'toasting' [applauding, giving recognition to] someone at a party or event, and is a traditional English public house ['pub'] salutation: when starting to drink you clink glasses and say "Cheers !". In neither case is a reply expected.