That's an excellant question,
Marky Lazer. Personally, I believe that Gandalf's staff, as well as the staves of the other wizards, were basically a tool. In the hands of the right person, it can be deadly, but with no skill to use it, it is useless. A wizard without a staff is not necessarily less powerful, but it is more difficult to transfer his power to reality(which would be called 'magic').
The staff is also a symbol, used to denote power and rank. The breaking of the staff is a way of saying someone has lost their rank or standing.
If you want to know more, try these:
Magic in Middle-Earth
Gandalf's Staff
Look especially at
Essex's post (#27) in
Gandalf's Staff. It's got a lot of good info.