Yes, you can.
Like most gaming systems, the Xbox 360 is made to play original Xbox games too.
try out Left 4 Dead Left 4 Dead 2 Avatar the last airbender any sport games army of two borderlands resident evil 5 gears of war 2
As of yet only Resident Evil 5 is available for the XBox 360.
I think you meant to say when your bored.... But I either play Black Ops, Dead Space, or Red Dead Redemption...and if you don't have any of thoses play Halo Reach it's one of the best games for xbox 360.
No the PS2 does not play Xbox 360 games only the Xbox 360 plays the games for it.