As fretje points out you can use the "cloud" for some things. Then there are services such as Dropbox and Windows Live Mesh that can help turn other types of files to sort of being in the cloud as well (Windows Live Mesh can do so with Macs as well).
Lifehacker runs articles about these types of things ever so often. A little searching and reading the comments might help you.
As for running applications off one computer while being by the other one, the short answer is No. The longer answer is that on Windows that is called Remote Desktop Services (not to be confused with RDP or remote viewing), formerly Terminal Services, and in today's jargon it's called VDI (Virtual Desktop Infastructure). You would need a server (computer able to handle multiple terminals), expensive software, etc. Then there is the question of how fast the whole shebang works. The major players in this space have just released products that can do full screen video. Real games with realistic frame rates? not happening, at least not yet. (There is a web company that was working on games running on their servers, haven't read anything about them in about a year so I can't give you a link.)
There are some web game portals, flash games, Quake Live, etc.
So it appears that your best bet is to use the cloud for as many SAAS applications as possible, use the cloud for file sharing and syncing, and try to use it for games as well. All is dandy until your Internet cuts out for an hour!
3Any preferred operating system for the question? – Gnoupi – 2010-02-12T09:39:54.247
Windows, if possible. – Enrico Tuvera Jr – 2010-02-12T15:28:07.527