"In dedicated server games, there is a server set up to run the game. This server is not a player, and often runs a different version of the game software. It doesn't need a video card, it doesn't need the game assets that are only needed for rendering. With a dedicated server, all players connect to that server - this is called a "star" topology. The dedicated server runs the simulation. What the player's computer sends to the server is something like: "I am moving in this direction, I am facing this direction, and I am shooting". The server updates the simulation and informs all of the players of the new state of things."
"In Peer to Peer games, the players connect directly to each other. One player's computer (we'll say Player A) sends packets directly to the other player's computer (lets call them Players B and C). There are two common forms here - the game may have a dedicated host player that all the players will connect to (you get A-B and A-C). This is similar to how dedicated servers work. One player is both the server and a player. A second form of peer to peer game is when all players connect to each other, and there is no dedicated host player. This is called a "mesh". Everyone runs the logic and then try to negotiate with each other. It is usually something like "I am here, facing in this direction, moving in this direction, I am shooting, and my shots are doing damage to this enemy".