Only one computer can connect to internet at a time!

Do I understand this correctly? You have one modem with one ethernet jack on it. Only one computer can have access to the internet at a time, depending on who you plug in to the modem? If the computers have wireless capability, you don't have a wireless access point or wireless router, so you can't take advantage of the computers' wireless. You don't have a wired hub or more than one ethernet jack on the modem nor do you have a stand-alone router (that plugs into the single ethernet jack on the modem) with multiple ethernet jacks, so multiple computers cannot connect to the internet simultaneously.

If this is the case, then when you turn on a computer that isn't plugged in, since there is no router there for it to obtain an address from (I'm assuming your computers are set up for DHCP), it is going to try to create/join it's own ad-hoc network and give itself a self-assigned address beginning with 169.x.x.x. (If you had a multi-port hub or wireless access point, with no routing capabilities, this is how you could set up an isolated network of computers without access to the outside world but with access to each other).

Meanwhile, your modem is either giving the plugged-in computer a 192.x.x.x or a 10.x.x.x address (unless you changed stuff in it) or it is passing your public IP address presented to it by your ISP straight through to your plugged-in computer (that was plugged when when you booted up).

So when you unplug one and plug in another, the network settings don't match. If you unplugged one and plugged in the other and then booted up the other one, I would expect to see a working internet connection at that point.

If my original assumption of your home configuration is in error, please describe how you are set up, e.g., internet<-->modem<-->router<-->computers or whatever. Maybe do that anyways, even if my assumption is accurate. Describe whether your modem employs NAT or is in bridge mode? Does it have a DHCP server and is it enabled or is it set up for static IP on your home LAN? Describe whether your router, if not built into the modem, employs NAT or is in bridge mode? Does it have a DHCP server and is it enabled or is it set up for static IP on your home LAN? How are your network settings configured in your computers? DHCP or other (what)? A visit to your modem and/or router configuration pages, to see what the external WAN address (a.k.a. public IP address) (at least the first one or two or three octets), and what do your computers' sys prefs IP address, router address, and DNS server addresses say they are, would be helpful, too (you can safely divulge publicly full 192.x.x.x, 10.x.x.x, and 169.x.x.x addresses because they are not routable). I don't know about other folks reading this, but I would need to know quite a bit more about your home network architecture and configuration.