'Throw up' would be the most commonly used. 'Vomit' is more clinical and formal. 'Puke' is considered more vulgar and usually would be used in spoken English, among people we aren't worried about offending. One slang term for this is 'toss your cookies".

All correct, but (just for the record) there is one more slang verb, "barf". Not as vulgar as "puke", more vulgar than "throw up" or "vomit", I think. :)