To exonerate someone is to declare him not guilty of criminal charges. This word is pretty much only used in reference to proceedings in a court of law. A word with a similar meaning that might be familiar is “acquit.”
When your next door neighbor was arrested for painting smiley faces on the front doors of all the houses on your block, evidence to exonerate him was discovered when the police found a young hooligan from the next street over with a basement full of paint canisters. The verb comes from the Latin exonerat-, meaning “freed from burden.” A criminal charge is certainly a burden, and when you're exonerated, you're freed from that burden.