When Anet released Factions, they released a factions-based (2 opposing sides) style of play into the game with additional PvP arenas, which they call Alliance Battles.
The 2 sides are Luxon and Kurzick.
These 2 sides fight against each other in Alliance battles which are 12vs12 people. No heroes or henchies are allowed here.
There is a border line which separates the 2 factions, and the line will move into Kurzick territory and back into Luxon territory. Wherever the line is will dictate which map of the Alliance Battles you will play on. As you go deeper, the maps become more difficult for the opposing side.
There are 2 maps on each side and one neutral map in the middle, making it a total of 5 maps.
There are Competitive Missions (8vs8) as well, which is Jade Quarry and Fort Aspenwood. Here the Luxons and Kurzicks also battle each other and must complete specific goals.
In all these battles you earn the relevant faction (Kurzick or Luxon) which you can trade for Amber or Jade with which you buy Elite armor or other armors.
You can also choose to donate your faction to your guild / alliance which will increase your title rank in that faction on your account and also increase the faction in the guild or you can buy pve skills / scrolls with it, of which both will also give you an increase in your title rank.
The PvE storyline of Factions splits at the Jade Sea / Forest where you must choose a side. You can play both but I would suggest playing one and then the other side. You cannot really maintain faction for both sides, you will lose one if you donate or buy something.
The storyline combines at Unwaking Waters again.
Most importantly, Guilds that are of the opposing faction cannot be in the same alliance.
An alliance is a grouping of 10 guilds of Luxon faction only or Kurzick faction only.
Alliances with high amounts of faction can own certain towns in the Jade Sea or the Forest areas.
I hope that covers the most of Kurzick / Luxon sides.