Yes and No.
You can either download the games off of xbox live and have them installed, or you can install it off the disc.
If installed off the disc you still need to use the disc to launch the game but once running it will use the hard drive to run, note that the disc cant be taken out at this time.
There's also "JTAG'ing" which is a modification to your console that allows you to run custom programs on your xbox some of these enable you to copy your discs to the hard drive as if you had downloaded them from xbox live.
However JTAG'ing may be illegal in your region depending on legislation.