What's stored on the 360 hard drive:
- The Dashboard application itself (typically)
- Any downloaded games from XBox Live (this includes demos, XBox Arcade, XBox Originals -- that is, original XBox games, which can be DVD-sized -- and XBox 360 titles -- again, DVD-sized)
- Downloaded content for games (such as Rock Band/Guitar Hero songs, Fallout/Oblivion add-ons, etc.)
- Saved games (which can be larger than you might think)
- Any ripped or downloaded music
- Any downloaded movies (unless you're using Netflix to stream them)
- Your profile (typically)
Now, as rlorenzo said, what you should get depends on what you want to do. My first XBox 360 shipped with the 20-gig drive, and I felt like I outgrew it pretty quickly, as I was getting Rock Band and Guitar Hero songs by the boatload, had most of the DLC for Oblivion and all of it for Fallout 3, etc.
When that XBox 360 died (out of warranty, sadly), I wound up buying a no-hard-drive model and a new 120-gig drive. I used the tools to transfer everything over and gave my old drive to a friend of mine. At 120-gig, I feel much more comfortable -- but am still not comfortable enough to load games from disc to the hard drive. (I'd prefer to have that space available for games I may download!)
If I were buying a new 360 today, I'd probably cough up for the 250-gig model. But that's because I know my usage pattern. You'll need to evaluate yours against the data that gets stored, and make the right choice for you.
(Also: There's nothing stopping you from using USB memory sticks in a 360 now, for all kinds of storage. Given that these sticks are dirt cheap, you could conceivably use a different stick for each game's specialized content, and go that route. However, the sticks are limited to 16-gig in size for 360 use. This may still prove more cost-effective in the long run, and perhaps more flexible as well.)