This article teaches you the many ways to increase faction standing in EVE Online. For a more general discussion of standings, please read How to Quickly Increase Standing. If you want to read more about corporation standing, read the corporation standing guide.
If you are doing missions for agents of NPC corporations, you will get an EVE mail from an agent who offers a mission that you can do for increased faction standing. This happens after doing 16 missions. If you successfully complete the faction mission, your standing with that faction will increase.
Your standing increases will get smaller as you do missions for the same agent. Be sure to move on after a while to a new agent so you can keep getting significant standing increases.
Train up your skills in Social and Connections. Doing so will increase your effective standing with all factions with whom you have a 0.0 standing or higher. If you have a negative standing with a faction, these skills will not increase your standing with that faction. If you have negative standing with a faction, train the Diplomacy skill to increase your effective standing with those factions.
Train the diplomacy skill to the highest possible level to increase your effective standing with factions with which you have a negative standing.
However, if your effective standing is still below -2.0 with a faction, even with the diplomacy skill trained to level 5, things are more difficult. If you can no longer work for a faction, find out if you can still work for an ally of that faction. For example, Minmatar and Gallente are allies. If you are able to work for Gallente but not Minmatar, then if you manage to boost your faction standing for Gallente, your Minmatar standing will also increase. Do 16 missions for an allied faction corporation to get offered an Important mission and get a standing increase with that ally, and with the faction that you have -2.0 or lower with.
You can find out which factions are allies by looking at the information window for a faction, clicking the Standings tab, and clicking Likes.
When you defeat NPC ships, you might find criminal NPC tags in the wreckage that can be turned in to the proper authorities. Read the NPC tags guide to learn how to reap the reward: a hefty faction standing increase.
The following systems have COSMOS agents for the specified factions.
The COSMOS agents in these systems give one-time-only missions that give faction increases. You can only do these missions once per character. If you fail one of these missions (and they are harder than the usual missions), you never get to try it again!