I have used DropBox for several years. It's basically a shared folder on your computer that syncs with the Dropbox servers. You can invite others to a folder and (if they accept the invite and install the Dropbox software on their computer) they will have a folder on their computer that mirrors the folder on your computer. Either of you can add, remove, change files in the folder and both of you will always have the latest versions of the files on each of your computers. Every time you save a file in the share folders it is uploaded in the background to the Dropbox servers.
It also stores past versions of files, so if you accidentally remove or change a file that stored on DropBox, you can easily recover it using the Dropbox web interface.
My only complaint is that there is no permissions control. You can share a folder or not, but you can't share a folder but not let people delete files. On the Mac (and probably Windows too) since your shared Dropbox folders are not actually "shared" folders from the OS perspective, when you drag files from your Dropbox folder it actually moves them instead of copying them. If you move them out a Dropbox folder then you have just deleted them from Dropbox. I've had to spend too much time undeleting files that someone accidentally deleted when they were trying to move them to their desktop.
Box.net I've heard is similar and allows more fine-grained permissions control. Also Google just launched Google Drive a few months ago which is very similar to Dropbox.