A lengthy walkthrough of the FATE system (below) goes into more detail. The large-scale open-ended events can be triggered regularly, randomly, or by talking to a particular NPC. You can find a FATE on your map, or just wander into an area with one. The video shows off two examples. In the first, the player joins up and battles a monster. In the second, the player needs to defend an area from monsters, with the promise of boss fights (and subsequent loot) if that phase is completed successfully. Square Enix is aiming to have roughly 300 of these included in the game when it launches.
The system adjusts variables like the number and difficulty of enemies dynamically as more players join, so a task won't suddenly turn easy-peasy if you get a bunch of extra players. That said, if you're too high a level, the game will let you know that you won't get rewards for participating. You can pick an option to artificially reduce your level and take away some of your powers, though. Check out the video to see it all in action.