Due to its volatile nature, any answer posted about Justice Points is likely to change quickly (at the very least with every expansion, sometimes with a patch).
A good place to look is Wowhead. It's officially supported by Blizzard and is considered the WoW bible by many.
Here is a list of ways to obtain JP (and ways to consume them as well): Justice Points
There are just way too many to list and the formating will probably drive me mad. I will just list a few ways you will get them once you hit Pandaria.
From patch 5.0 or maybe an earlier patch, you have to be level 70+ to start earning Justice Points.
- Random regular dungeons (not available to level 90s) - 60 points per dungeon run, no limit
- Heroic dungeon bosses - 100 points per boss, no limit
- Normal scenarios - 50 points contained in Greater Cache of Treasures per run, not limit
You can check the amount of points you have by opening up your character page, and clicking on the currency tab.