If you get a router, you should be only limited by your bandwidth. The more people you connect to a connection, the slower it will get (assuming the connected users will use the connection simultaneously). But the physical number of computers you could connect is quite high. For instance, your setup could be like this:
Internet---Router---Computers
Where the number of computers connecting to the internet would be limited by the number of ports on the router
Or
Internet---Router---Switch---Computers
Where the number of computers would be limited by the number of available ports on the switch in addition to the number of ports on the router. You could plug multiple switches into one another, such as:
Internet---Router---Switch---Switch---Switch
Computers would be able to plug into each switch. So, the number is actually quite limitless.
The default for most routers will translate your Public IP address to a Class C internal IP address. This is done through Network Address Translation. You could use a class B or a class A network to connect more PC's than in a Class C network, but since you'll be able to connect 253 computers to a Class C network (254 actually, 253 computers and 1 router) I doubt you need detail about changing to a different network address.
Hope that's what you're looking for...
deeno