A game server (also sometimes referred to as a host) is a server which is the authoritative source of events in a multiplayer video game. The server transmits enough data about its internal state to allow its connected clients to maintain their own accurate version of the game world for display to players. They also receive and process each player's input.
Dedicated servers simulate game worlds without supporting direct input or output, except that required for their administration. Players must connect to the server with separate client programs in order to see and interact with the game.
The foremost advantage of dedicated servers is their suitability for hosting in professional data centers, with all of the reliability and performance benefits that entails. Remote hosting also eliminates the low-latency advantage that would otherwise be held by any player who hosts and connects to a server from the same machine or local network.
Dedicated servers cost money to run, however. Cost are sometimes met by a game's developers (particularly on consoles) and sometimes by clan groups, but in either case, the public is reliant on third parties providing servers to connect to. For this reason, most games which use dedicated servers also provide listen server support.
Listen servers run in the same process as a game client. They otherwise function like dedicated servers, but typically have the disadvantage of having to communicate with remote players over the residential internet connection of the hosting player. Residential connections rarely support the upload requirements of games with many players; the typical limit is 16. Performance is also reduced by the simple fact that the machine running the server is also generating an output image. Furthermore, listen servers grant anyone playing on them directly a large latency advantage over other players ("host with most") and cease to exist when that player leaves the game.
But listen servers have the advantage of being essentially free and not requiring any special infrastructure or forward planning to set up, which makes them common at LAN parties where latency and bandwidth issues are not a concern. They are also common in console games.