Game server hosting is a specialized type of web hosting specifically for online games. Game server hosts offer a range of services from shared servers to manage dedicated servers. There is nothing preventing someone from getting a standard dedicated server and running a game server on it, but companies specializing in game servers tailor there setups to the needs of gamers and have the expertise to help when help is needed.

Selecting the right host

• Ping
  The most important thing for gamers is ping. 'Ping' is the amount of time it takes for a piece of data to get from your computer to the server your interacting with. This is mostly determined by the number of servers your data goes through (traceroute) before making it to it's destination. A high ping means that there will be a significant delay between an event happening in the game and you seeing it on your screen. Someone with a lower ping will have a huge advantage because they will always be one step ahead. This means that picking a server with low ping for you and your friends is very important for a gaming server. Ping is primarily determined by physical distance so picking a server that is physically close to you is important. A good technical explanation of how this works and why it's important, along with some other gaming info, can be found here.
  A good game host will offer a way for you to test the ping before signing up.
• Everything Else
  Again, ping is the single most important thing for gamers. Everything else is secondary to ping. Once that is out of the way it is time to look at the cost and features of different hosts. The range of these varies greatly from host to host. You can get everything from a shared server to a fully managed server that only requires you to play and have fun. The price and features offered change frequently so shop around.