The meaning of vomit, puke and throw up are essentially the same, only retching is a bit different. To retch means you might have a big stomach spasm but don't actually puke anything up, typically happens after you have puked. The others definitely mean you are regurgitating what you have eaten or drunk recently. However the usage is a bit different. I would say vomit and throw up are much more formal than puke.

and vomit is what a doctor or medical worker would use in a medical situation "If the child ingests poison, you can induce vomiting to get it out of his system." throw up is what most people use most of the time "I was sick with the flu and threw up all night long." puke is much more informal. Young people use that more often, I think. "Dude, after last night's party I thought I was going to puke my brains out." : )

Oh Ukk, what a horrible subject!! :D Vomit - puke and throw up are what we do when bringing up food from the stomach - Retch is different, we retch after being sick and there is nothing left to bring up. Also, we may retch from a nasty smell or sight, it's as if we are going to be sick, it comes from the stomach but it's dry.