If you've been called zany, you are goofy, wacky, and clownish. Zany describes very silly people and behaviors. If you break into a bad, old-guy imitation of hip-hop, you might be trying too hard to be zany.
There's an old character in comedies from the 15th through the 19th centuries who always had the Italian name Gianni, or Giovanni, another form of which was "Zanni" — from which we get the adjective zany. Just as there are good clowns and kind of scary, weird clowns, zany describes both truly funny and laughable people and things, as well as foolish, or ludicrous, attempts at being funny. Something zany makes people laugh, unless it's weird zany and just makes them cringe.