Vomiting is the act of forcefully expelling the stomach's contents out through the mouth. It's usually an involuntary action brought on by nausea, but can be deliberately provoked. Almost everyone has had this unpleasant experience at least once (if not many times) in their life.
Nausea refers to the unpleasant "queasy" feeling that you get when you're about to vomit, one that's accompanied by uncontrollable contractions of the stomach that begin before and continue during vomiting.
Nausea and vomiting aren't normally medical conditions themselves, but are commonly symptoms of a disorder or illness.