Some people already wrote what could be considered an answer, but I'll try and distil what they said to an answer:
Dark Souls has an master server, probably one for every region, that is responsible for hooking you up with other players.
So when you put up a summon sign, you tell the master server that a player may join your world.
However, when that player has joined your world, either your machine or the players machine will be the player server.
The player that serves as the host to the game will have all commands from the other players being hosted relayed to its machine, which in turn will sync up the other machines to the current state of the game (unless they use p2p, in which case all machines will talk to each other, and wont have to relay their message through an intermediary host machine).
So what they mean when they're saying that they will shut down the servers is that you wont be able to find players to join any more.
Some games, like Gears 3, have dedicated servers, that means that in addition to providing a master server, it also serves the game, so that no player gets an unfair advantage out of being the host for the game.
This is all speculation, I don't know if they actually have this setup, but I suspect it. I've done some shallow googling, and it seems that it is p2p (so no centralized server), and my theory of a master server seems to be the case, but I am not 100% sure.