What is a good bike for college? I'm looking for a good bike for college. I need to be able to go across paved roads, brick roads, cut across grass- and possibly jump down 5-6 stairs. 

This video is from the school I plan to attend:


                
            

Any suggestions would be greatly appreciated. My price range is $400-$600

Do I need a road bike? Dual sport? Mountain bike?
The most important thing for a college campus (and commuting in general in places where there are a lot of potentially not nice people) is to have a cheap bike and locking it properly - in particular, one that doesn't attract attention. Old and scuffed up bikes are best for this. I think an old mountain bike or flat bar hybrid (like a 80s/early 90s specialized hardrock/rockhopper or an old giant boulder is ideal) with a rack attached are pretty good options, along with some slick tires and lights and a helmet. The total for this should be around 210 dollars (lets say 150 for the bike + any work it needs done like derailleur adjustments, possibly new tires, etc., a helmet for 30 and lights for 30). Add 30 if you want a cheap rack, and another 40 or so for a decent lock (say, a Kryptonite evo mini, or if you're paranoid spend a bit more and get a Kryptonitefahgettiboutit). A U-lock is recommended - chain locks give you a bit more diversity for locking things, but they're too heavy to carry around for good ones (and most campuses which are bike friendly have plenty of places where you can ulock to). Fenders/mudguards may also be a decent investment for rain (though college students are a mess anyway, so I don't think it makes much difference). I don't think a road bike is necessarily a great idea given the short distances involved and the not necessarily useful geometry for commuting on most of them. However, if you do find one which you like, take it out for a spin and see how it does. 

Ask around the community and see whats there for used bike shops and try to find a decent bike there - you'll also probably get some help on fit there as well. Jumping down stairs depends highly on the type of stairs, but it is too reckless behavior for most college campuses. If you are hitting a lot of bumps while commuting, a rigid bike with fat smooth tires or one with front suspension may be helpful (though, I think for 99.99% of commuters, rigid is the way to go). Stay away from full suspension (which you will only find on the Walmart level of bikes). Also, it is good to have non-quick release wheels + seat post.