Quote: " But I thought that the client computer must be connected to the same IP address as the one the game is hosted on"
I'm not sure exactly what you mean, but this is how it works:
The host 'hosts' using its IP(if you are directly connected to the net this is your internet IP, but if you are on a LAN then it's your computers LAN address), and others connect by specifying the internet IP of the host(unless you are both on a LAN, in which case you use their computer's LAN IP).
Of course, over the net you will have to set up your router's port forwarding..