Corporations are analogous to a guild or a clan. Each player must be in exactly one corporation. Player-made corporations can declare war on each other and their members can also freely fight amongst themselves.
In the Eve universe, there are two primary kinds of corporations. The first are NPC corporations, which exist primarily as a starting point for new players and the infrastructure of the economy of the game. The second kind of corporation is the player created and owned corporation. This is the exciting part. With player corporations, players can become part of independent economic and military entities in the Eve World!
These corps run the gamut from the simple roving pirate corps, to the common industrial corps, and everything in between.
For information on starting a corporation, reference the Starting A Corporation guide.
Among NPC corps, there are a number which are used as starting corps for players, and others are more specialized.