Corporations take up numerous business models such as mining, manufacturing or "ratting" (hunting NPC pirates for their bounties and loot). Corporations can levy income taxes on their members, which skim off a percentage of every member's earnings. Many corporations offer a variety of benefits to their members, such as free or discounted ships, equipment, formal training, and organized corporate group operations.
NPC Corporation (Non-player Corporations) are run by NPC's rather than actual players. They offer a number of services such as refining minerals, manufacturing, mission-running and more.
For your loyalty in running missions for an NPC corporation, they will pay you ISK and may offer your bonus rewards. Loyalty Points are also earned which can be exchanged for items at a Loyalty Store for that corporation to exchange for ships or items.
Pilots who belong to an NPC corporation are immune to war declarations but are still vulnerable targets to other acts, such as piracy.