Network and proxy server questions

Does a proxy server go between clients & Internet/network, or anywhere on the network? Can you require use of a proxy server without installing some sort of client on each PC?
I see the checkbox on IE, Use Proxy Server: & the address and different ports for Web, FTP, SSL, etc. Is it possible to force clients use that, and if they don’t – no Internet access for them? Seems too easy to just remove the check box or would you deny ports 80,21,443 access to all but the proxy server? Thanks!