Well here's some pro's and cons.
Desktop
Pros: If you build you get everything you need and everything you don't, you get to personalize your PC to your usage.
It's better bang for buck than a laptop.
No spam on pc.
Better gaming than a laptop in the price range.
Cons: Not portable.
The gaming pc market is changing so quick a $800 pc today will be a $500 in 2 years.
Have to buy monitor if you don't have one, same goes for mouse, keyboard and an OS.
If you choose this, here's a nice build:
http://pcpartpicker.com/p/1Ww9N Now I listed this myself, and it does go $34 over your budget, but this rig will serve you well. PS: I listed it without an OS as most people already have a spare OS laying around.
Laptop:
Pros: Buy it and play, no assembly needed.
Portable.
Comes with an OS.
Cons:
Not as powerful.
Comes with crap on it.
This may not be an issue for you, but most new laptops come with windows 8 on them.
Hope this helped you, ask me if you need anything else.
Oh and whatever you do, if you go with laptop do NOT get an alienware.
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