1) Basically anything with a "complete" checkbox, such as Quests and Leves
2) FATEs (every member contributes their own points), Treasure maps (points are just from the one instance of the map, not per person)
3) Gathering HQ items
4) Crafting HQ items

All of these generate points. More for higher level activities. Gathering/crafting is in pretty small amounts.
Also if less people are online, slightly more points are generated per activity.

In the near(?) future (possibly 2.2), Duty Finder activities and PvP will also generate points.