MMORPG.com EVE Online Correspondent Sam Guss writes this article giving players a bit of an overview of Loyalty Points in EVE Online, with a focus on some of the NPC factions you might run into.

It's very easy in the game of Eve Online to find an agent, talk to them, get a mission, do the mission and get a reward. This reward in the beginning is usually ISK and sometimes loot in the form of an implant, a ship module, a blueprint or even a ship. As you move along in Eve Online though, these rewards turn more into ISK, LP (Loyalty Points) and Faction Standing.

Players of Eve Online will recognize LP's for faction ships, faction modules and implants. These are after all some of the most sought after items. For a new player, the most common thing to happen if you are in a good corporation is to run higher level missions with more experienced players. This results in more ISK, an introduction into LP and of course standings – both for a particular faction, as well as individual NPC corporations. This does have one drawback however and that is it only exposes you (usually) to one or two NPC corporations, usually the major ones for a particular faction.

Read the EVE Online Loyalty and Factions Article

 Very good explanation, although some info is missing, I think it reflects the reality of agent business quite well.

I would add that you can train some social skills that let you gain faction/agent standing, and that doing missions for one corporations gives not only pluses for that corporation, but minuses againts other corporations involved in that mission plot or simply opposed to the corporation youjust worked for. 

Good article on how to choose agents to work for and I agree that it does matter a lot if you roleplay, e.g. as gallente pilot would never work for caldari agent in EVE, never ever for sure :)

I would recommend for a new mission runner to train some social skills as soon as possible and start by training Social to 1, then Negotiation to 3 as it gives 5% more pay per level and that is +15% more ISK,  and then Social to 3 and after that Connections to 3 at least.

That way you can skip slaving forever working for level 1 agent at low pay and go directly to level 2 agents when you can fly Cruiser effectively. Later when you can fly Battlecruiser you can train Connections to 3 to double your ISK reward and LP from your agent. For level 4 missions you need Battleship.

Have fun and be ready to loose a ship or two in your mission running. I did loose a few ships at first but lately I can solo all level 4 missions in my fine Hyperion Battleship and has not lost a ship for half a year. Good luck.

Faction items are nice, except that they are too expensive for most of us to chance losing them.  Because of that they are more like trophies, you can display them, but dare not use them.

Remember when a faction ship is lost, it is lost or good and the nature of this game is pvp, losing ships is not uncommon.  Same goes for loading a ship up with faction items, just makes you more of a target.

So, the rewards, while nice, are not that useful to the average player.  Of course the wealthy ones don't care.