Faction Warfare involves PvP in the faction-contested lowsec regions. The four factions gain and lose territory as they fight, and participants earn rewards based on their efforts to help conquer territory for their faction.
Faction Warfare can be a great way to find semi-structured PvP objectives with potential rewards. Because faction warfare sites have restrictions on what classes of ships can enter them, you have the option of fighting with only smaller classes of ships that are often cheaper to fit - which means you can fit (and lose) more of them as necessary.
Faction Warfare rewards involve significant amounts of Loyalty Points (LP) which can be exchanged for military faction equipment, either to use yourself or sell on the market for ISK.