Favor is your current relationship-status with a certain NPC. By raising your favor with that person you'll recieve various rewards such as the ability to utilize new skills not available elsewhere, storage space in their inventory, experience, recipes, ability to list a limited amount of items for sale to other players (similar to auction houses in other MMO's) and increased value on items sold.
Favor is raised by giving specific gifts to NPC's, doing favors for them or completing quests. By reaching 100% favor you'll take the relationship to the next stage, the amount of favor gained has diminishing returns based on how many times you've reached 100% favor. The different stages of relationship you can achieve with an NPC goes as follows:
Despised ⇐ Hated ⇐ Disliked ⇐ Tolerated ⇐ Neutral (starting) ⇒ Comfortable ⇒ Friends ⇒ Close Friends ⇒ Best Friends ⇒ Like Family ⇒ Soul Mate
To get from Neutral to Comfortable requires 100 favor.
To get from Comfortable to Friends requires 200 favor.
To get from Friends to Close Friends requires 300 favor.
To get from Close Friends to Best Friends requires 600 favor.
To get from Best Friends to Like Family requires 800 favor.
To get from Like Family to Soul Mates requires 1000 favor.
NPCs in are a cool bunch and most of them want to "hang out" with you! They will reward you with Favor as well as miscellaneous rewards in theme with the activity you selected (typically an item and/or Councils along with some sort of XP.
You don't need to remember to do that every time you log off by the way. You can chose an "Hang Out" activity with any NPC as you play, and the last one you selected will be active once you log out. NOTE:You also don't have to worry about logging back in sooner (whew), the activity will continue next time you log out.