DropBox is one of those "cloud" places. In the simplest terms, it is a remote storage facility for your "stuff". Back in the "good old days", you had to have your data and your application in the same place which was on your hard drive. But with networks and now the Internet, your "stuff" can be someplace else, anyplace else and all you need is access to it in order to use it.
Think of it like your ATM or debit card from your bank. It used to be that you used to have to carry cash money in your pocket in order to buy something. But now with most businesses taking debit cards, there is no need to carry your physical dollar bills with you. All you need is access to your money and that's what the debit card does.
Probably one of the biggest advantages of the "cloud" is that it makes it very easy to share your "stuff" with other people as long as you give them access to it. Also, if you need access to files at work and at home, you always know where they are.
Clear as mud, eh?
As far as MobileMe, I have never used that so I don't know how they compare.