A dedicated server is a storage centre for data, essentially.
To understand how it's a part of a game, you first have to understand its other applications. Basically, whenever you download something from a website, you are accessing the server (or location, if you will) in which the data resides. Your computer is taking the data from that place and saving it either temporarily or permanently.
A dedicated server in a game is a server that's dedicated to processing the game data. That means that when you perform an action, you send data to the server and then the server sends that data to the other players. Likewise, you're receiving data about the actions of other players. The time this takes is based on your "ping" or "latency". A latency of 1000 means there's on second error from when data is transmitted to when you receive it. Generally, people like to play with a latency of about 100 or less. Ideally, you're in a server with about 10-50 latency, wherein the difference between data being transmitted and you receiving the data is minimal.
Dedicated servers are awesome. Here's why:
- You get to know a bunch of players who play in the same location and same timezone as you.
- Clans can buy or rent servers for their own use
- Privately owned servers can be customised, taking advantage of mods and custom maps
- Competitive dedicated servers can have passwords, meaning that only the specific combatants can enter
In essence, dedicated servers put the multiplayer experience in the hands of the players themselves. A game with a good community, dedicated servers and players that keep coming back will thrive for a long time.