What can I do with them?
Gorble Gibbernof [ Moderator ]
Loyalty points are accumulated by doing missions for agents of one corporation over time. The higher the level the more LP you get per mission, also the higher you effective standing with the agent the more LP you get.
Each corporation has their own pool of rewards for you to choose from and these can be viewed in the LP store previous to doing any missions for that corporation. Also the rewards depend on what kind of corporation. For example the nation corporations give out Navy ships and ship modules as well as skill hardwiring. Most non nation corporations give out implants for both attributes and skills. Pirates give out implants, ships, and ship mods.
A searchable database of rewards can be found here: http://joonicks.eu/eve/lpstore