Almost all stations in EVE are owned by one NPC-corporation or another, and the agents located on those stations always belong to the corporation that owns the station (with a few, very rare exceptions).
Some of those agents are willing to talk to you and give you missions. If you accept and complete such a mission you are rewarded with ISK, Loyalty Points (LP) to that agent and with a Standing-gain towards that agent and the corporation he works for (If you accept, but fail, a mission you get a Standing-loss towards that agent and his corporation & faction).
More on the topic of accepting missions
Since Revelations 2, a new concept called the Loyalty Store has arrived. Dock at a station owned by the corporation you are doing missions for and check out the offers. They all vary a bit.
Your character gains Standing (both positive and negative) towards three thing in EVE: Agents, Corporations and Factions. Standing goes from -10 (they hate you) to +10 (they love you).
When you are doing a regular mission for an agent, you get a Standing-gain towards the agent and the agent's corporation if you complete the mission successfully. If you fail the mission, you get a Standing-loss instead. You typically lose around 4-5 times what you would have gained if you had completed the mission successfully.
Since Revelations 2, it has become possible to do missions in a group and share the benefits, both ISK, LP and standing. This requires the use of a gang and the mission "owner" to use the "Me and my gang has completed this mission" option when completing the mission. This is a nice way to help new players gain standing and thereby access to better agents - and a nice way to reward participation in group-style gaming, which is one of the best things Eve has to offer.
Agents are defined by their level (1-4) and their Quality (from -19 to +20). The level of the agent corresponds to the level of the missions they are offering. Level 1 missions are the easiest, but also gives the least reward in terms of ISK, LP and Standing. Level 4 missions are very hard but gives great rewards in ISK, LP and Standing.
The agents Quality shows how much ISK, LP and Standing you gain from him (within the level). The lowest quality agents give the most crappy rewards while those with the highest quality gives the best. Note that an agent of low quality, but higher level still gives better rewards than an agent of high quality but lower level.
The better Standing you have with an agent, the better his *effective* quality is. An agents real quality might be very low, but if you have a high Standing with him, then he'll give you rewards as if he was of a much better quality (but still of the same level).
Note that different missions give different rewards depending on their difficulty. A hard combat-mission gives more than an easy courier-mission. There is also a random element, so the same mission does not give the same reward each time you successfully complete it.
When you gain (or loose) Standing with an agent you also gain (or loose) Standing with that agents corporation. Standing towards a corporation determines some important thing in EVE:
If you want to know what agents from a given corporation is willing to talk to you, then look at the corporations information (there is several ways to do that) and look at the agents tab. Under the different types of agents you can see who is 'Available to You' and who is 'Not Available to you'. The different types of agents determine the frequency of what mission-types is offered (Combat- or courier-missions).
Some ways to look at corporation info:
Every this you have done about 16 missions from agents of the same level, belonging to corporations that belongs to the same faction, you'll get a mail about an 'Important Storyline Mission'. Fly to the agent the mail is from and do his mission. You get the normal rewards and the normal Standing-gain towards the agent and his corporation - But you also get a Standing-gain towards to the faction the agent is from.
Faction Standing allows you to work with more corporations within the same faction. If you haven't done any missions with a corporation, your faction Standing substitutes for corporation Standing, so you don't have to start with the lowest level and quality agents if you want to work for a new corporation within a faction. Also: If you have a negative Standing of -2.0 or worse towards a faction, no agents belonging to corporations within that faction is willing to talk to you.
The different factions in EVE likes or dislikes each other, and if you gain Standing with one faction you also gain Standing with those factions that likes the faction in question. But you'll loose Standing with those factions that dislikes your faction (they don't like people that ass-lick their enemies…).
Another way of losing Standing with a faction is to kill ships that belongs to corporations within that faction. You'll soon notice a sharp drop in Standings towards the pirate-factions - They take a dim view on you when you kill their ships in missions or ratting, for some reason… More important is, that some missions forces you to kill ships belonging to corporations in the four 'player-factions' (Amarr, Caldari, Gallente and Minmatar). So if you want to run missions from corporations within all the big factions, you should think carefully about accepting such missions.
Important: If your Standing with a faction drops to -5.0 or lower, you become Kill-on-Sight (KOS) from NPC-ships belonging to the navy-corporation of that faction, as long as you are in that factions area of space. This might make it harder to move around in the EVE-universe, but some players is hardcore and just tank their way through.
There are some skills worth looking at when it comes to Standing:
