Prop trading or propriety trading is trading the capital of the institution itself. Basically say goldman has 500 million in revenues. What it would do is have the prop traders trade that 500 million for gains. Nowadays, in the post web boom era, prop trading is making the most profit for ibanks. (not sure but i think its like 30-40% of the income). Prop traders work with the rhythm of the market and usually employ tech analysis but some use fundamental analysis for the more long term positions. You have t show that you have skill in trading. Let me put it this way......education isnt the top 5 most important criteria in trading. The field of trading is much more level than that in M&A whom I really consider to be prestige whores. Prop traders usually have some experience but some make it in due to their experience.
Hedge funds are like prop trading firms but they take other people's money. Think mutual fund with a twist. Currently hedge funds have very few restrictions so they can probably employ just about every trading strategy you can possible think of. However, this will probably change as the SEC are blaming hedge funds for some of the problems in the economy now. Hedge fund traders apply a lot of arbitrage positions. They take advantage of inefficiencies. Thats why many hedge funds are based in latin america, europe, or asian positions. Lots of bond arbitrage and mergers arbitrage is also used. To break into hedge funds out of school you should definately be at the top schools. Think Chicago, Harvard, Wharton and Michigan. Actually thats it. Most traders come from those schools. Hedge funds like computer majors and math majors because arbitraging is HIGHLY quantitave. However, economics is very well represented.
To sum it all up if you are good at trading you will love prop and hedge funds because hours are better, less of an hierarchy and more excitement. But if you aren't good and think you cant withstand the pressure stick to slave work....errr.....i mean corp finance.
Good luck.