Peer to Peer: This is a multiplayer architecture where everything is done client side and then clients simply update each other to keep the game in sync. Examples: Civ4, Starcraft, Diablo.
Client/Server: A, normally dedicated, server controls the flow of the more important aspects of the game, ensuring a balanced atmosphere and a world that cannot (unless poorly coded) go out of sync. Because of this, the server requires significantly more upstream bandwidth than any of the clients do. Due to the upstream bandwidth requirement, most servers in this architecture are either installed covertly within other companies (used to happen a lot at ISPs) or leased from small/medium sized server companies. Either way, (regardless of whether you've ever personally paid a cent towards servers) the end user is relied on to host the game servers in this architecture. Examples: Quake, Unreal Tournament, Call of Duty.
Client/Multi-Servers: Best way I could classify this. About the only time this scenario occurs is within the MMO genre. Because of the size of the world and number of clients, the game world has to be engineered to run on a large number of servers all having their own specialized functions. Database servers, logon servers, zone servers, etc. MMOs often have backups as well. Because of all the high end infrastructure, there is a fee associated with playing the game. Examples: World of Warcraft, Everquest.