Loyalty Points are how corporations repay members for services rendered. One example is given through missioning, as an additional reward for completing them. Other way of getting loyalty points is Faction Warfare, which gives you LP for that factions militia, and Incursions, which give Concord LP.
LP is spent buying items specific to that faction or group. Each race has things they specialize in, and so their faction items will be better (ie. Caldari Navy Ballistic Control System). LP is locked to the corp that you received it for, with the exception of Concord which can be traded to any factions LP. You can check each corps LP store by going to a station owned by that npc corp and checking their Loyalty market.
Some items are generic and can be found in nearly any LP store, such as implants.