Hypothetically speaking, let's say you live in a really big house and have 7 computers (designated them A-G for simplicity), each in different rooms/spaces of the house, connected either by wire or wireless to a router and sharing a single internet connection. Is there any way to turn one computer (A) into a server that'll run all my programs/games and store all my files on it, and the rest (B-G) will simply be "terminals" that allow me to access stuff?
Dropbox is a great cross-platform way to sync up to 2GB of content for free. It will exist in the cloud (it's Amazon's S3 at the back end) and sync to your Windows and Macs. The latest versions include "LAN Sync" which syncs the data across your local network instead of up to and down from the cloud everytime. I use it for a subset of my data.
The rest of my data lives on a server (Server 2008 R2) and is accessed over the network. The "Documents" directory on my Windows 7 PCs is redirect to a share on that server. My MacBook Pro can access the same data, and my Windows 7 laptop uses Offline Files to ensure it's all available when I'm not on the LAN.
My pictures, video and music are also shared on the network from the main server; although I don't take them with me with Offline files (too much content/don't need it on the road). My iTunes folder is on the music share which means I can open iTunes to the same library on multiple machines - just not at the same time.
There are many, many ways to skin this particular cat. This is just my (Microsoft-centric) take on it.
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!
If you are going to be doing heavy gaming, it's going to be tricky to get good performance on the client PCs unless they have good hardware of their own.
You may want to consider having the server be simply a large file server, running an array or two of Raid 5 disks, and then limit gaming to only a few PCs to save on expense.
A simple (but not elegant) solution would to have the server store Virtualbox disk images, then have the PCs run a light-weight OS just for the purpose of hosting the virtualbox guests stored on the server.
Very trivial to keep (everything) in synch, but a tedious process saving and loading the image across the network whenever you change computers. (Unless you have some sort of gigabit lan set up).
In my experience there is no better solution than Dropbox, where update is lightening fast. In free account you'll be able to use it only on 2 computers.
The 2nd option would be Sugarsync, where same 2GB in free acc, but there is chance to select your own folder for sync. But update falls behind Dropbox (considering an pdf file 30 MB or more).
May be use these 2 free sevice, or check plans at Dropbox.
To add: i've used webOS, like Glide or g.ho.st, both give ca. 30 GB, but upload is a pain (on a broadband of 3mbps)