Minecraft servers are multiplayer games that you can play with your friends, meet new friends, or maybe even meet new enemies. For many players, in an un-modded battle, Multiplayer servers beats singleplayer. There are two primary types of servers. They are:

Vanilla Servers - These servers don't get much publicity, but they are easy to make. If you plan to make a server for just you and your friend to play on, then this one is best.

Bukkit Servers - 24/7 Portfowarded Bukkit servers are often the most popular servers. Using something like Hamachi will limit your players (and prevent new ones), but Portfowarding allows players to join as long as the server is up and not whitelisted.