A game server is a specialised computer system that's sole purpose is to process the code required to make multiplayer games function; process player position data, perform physics calculations, hit registration, synchronisation and more... They are essential in providing smooth gameplay that is expected by the modern gamer. Without them, multiplayer gaming wouldn't be as fun, and definitely wouldn't have grown to the scale it has.
To elaborate, a game server determines the game state, then sends that to all players, it calculates the game state every tick, and tick rate is therefore the measure of how often the game state is calculated. Game state is a snapshot of everything occurring in the game world, positional data, physics calculations, environment data, scores, etc... Tick rate is a measure of game state updates. Tick rate is the number of times the game state is calculated and then broadcast to each connected player, every second. The higher the tick rate, the more accurate the game is. Why does tick rate matter? Well, you always want the game you're playing to be accurate, best example is a gun fight, a low tick rate can cause shots that should hit your target, to miss. This leads into why server hardware is so important; server tick rate is directly supported by the underlying hardware, powerful hardware leads to a high tick rate and vice versa. This is because the more often the game state needs updating (a tick), the more CPU power is required to do so, updating the game state 128 times in a second is a damn sight harder than 32 times in a second...
Game servers use the client/server model - simply explained, the server processes almost all the necessary data, then sends it to any connected clients (players). The alternative is the peer-to-peer hosting model, this is where one client (peer) is the host (usually the first to start the game, or the client with the strongest network connection), all other clients send their data to the host, the host then does the processing and sends data back. The client/server model is superior to the peer-to-peer model for many reasons...
The biggest advantage of client/server model game server hosting is accessibility; if the server is online (which it will be, 24/7/365), then any client can connect and enjoy the game. With peer-to-peer hosting, you need the host to leave their personal computer online 24/7/365, that is not very often not feasible. You can get around this by sending the relevant save data at the end of a play session to the next person that will play, but that's time consuming, tedious, and difficult in comparison to having a game server just host the data all the time.
The other big advantage of client/server model game server hosting is superior performance; due to the way peer-to-peer hosting works (data going back and forth between all clients) when a client with poor hardware and/or network connection connects, everyone is held back by them, this doesn't occur with client/server hosting as the server processes' data for each client asynchronously (when data is received, its processed and sent back, theres no waiting for another client).
Game servers, VPS (Virtual Private Servers) and Dedicated Servers all use the client/server model for the aforementioned reasons
Game servers are typically run on shared hardware (this provides cost effectiveness compared to a VPS or Dedicated Server) which means that multiple customers game servers are powered by the same underlying hardware. The CPU, RAM and Storage is split between all the customers. Obviously it's in the best interests of the hosting provider to manage the amount of customers game servers on the underlying hardware to ensure optimal performance. This is very difficult as each game server uses resources differently.
In terms of the actual hardware used for game server hosting, it's not dissimilar to that of a gaming PC. This is because at it's core, every game is a big loop of events (process player position data, perform physics calculations, hit registration, synchronisation and more) that must be performed one after another, cannot be performed at the same time. That therefore favors fast single core performance (high clock cycle speeds). Game servers are no different, they not coincidentally are also a big loop of events, and therefore also benefit from fast single core performance. So using traditional server CPUs that have a large amount of cores, but each core is relatively slow, does not provide the best performance. You will get much better performance from a game server when using gaming PC hardware that has a lesser amount of cores, but each core is relatively fast - high clock speed and high instructions per clock cycle (IPC)