Standings are shown on the Interactions tab of your character window (same window that has your skills). You can see all of the factions, corporations, and agents with whom you have standings. When you do an agent mission, the agent gives you a modest standing increase, and their parent corporation gives you a derived, smaller increase. If you do storyline missions, the parent faction also gives an increase, factions friendly to the parent faction get a derived increase, and factions opposed give a derived decrease. Training the Social skill will up the amount of standings change for each mission.
Obligatory EVE Uni wiki link: NPC standings - EVE University Wiki
More details on standings and how to effectively manage them can be found from the pages related to the FACTION STANDING REPAIR PLAN aka ‘The Plan’ and the website of the USIA service.