A normally formatted memory stick (not one for save files) can be used to play music/videos etc. stored on it. You can't copy files to the Xbox, but you can use them from it.
You can't add to the Xbox formatted filesystem from a computer.
The answer to your particular situation is "kind of." You can certainly hook up a 1TB hard drive via USB 2.0. The gotchas are:
The Xbox 360 doesn't support USB 3.0
The Xbox 360 currently only allows for 32GB of useful storage per USB device (UPDATE: This is changing in 2015, and 2TB drives may be fully compatible in the future, see here for details)
To put that 32GB limit into perspective, downloaded games can range from 500MB to upwards of 5GB, depending on the game. You can also install games from disc onto the hard drive or external USB storage, and those tend to run around 6GB per disc installed, give or take.
However, if you plan on using this terabyte drive exclusively for Xbox 360 content, most of it is going to go to waste. It seems like Microsoft's intent with adding USB storage support was to replace the old Xbox 360 "memory card" with the option for using a USB flash drive, as they hint at in the FAQ section of this help page.
These issues may drive you to purchase an "internal" Xbox 360 hard drive, which I believe uses SATA (instead of USB) and is not limited to 32GB.
Best Answer
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