Introduction to the Legacy System
The Legacy System, once fully implemented, promises to be an interesting addition to Star Wars: The Old Republic. According to James Ohlen and this blog post on the official site the legacy system "will grant your characters access to powers, objects and other cool benefits." The legacy system is unlocked for your characters upon completing Act 1 of a character's class quest. This comes immediately after you complete your quests on Alderaan. Unlocking your Legacy allows you to choose a legacy name, and grants all your characters the ability to begin earning Legacy experience and gaining Legacy levels. The higher your legacy level, the more benefits you'll have unlocked through the legacy system.

Gaining Legacy Experience
Legacy experience is earned whenever you earn normal experience. The amount of legacy experience gained is calculated by multiplying your normal experience gain by a multiplier. Refer to Table A below to determine what multiplier to use, based on your character level and the source of the experience gain. For example, if your character is level 50 and you complete a quest -- let's say, [WEEKLY] Blacken the Ice -- the multiplier would be 0.2292796. That quest rewards 14625 experience. 14625 * 0.22928 is 3353 legacy xp. If you were playing a level 15 alt completing a level 15 quest like Buried Power would earn you 4430 normal experience and 0.264063 * 4430 is 1170 legacy XP.

Maximizing Your Legacy Leveling
Efficiently gaining legacy levels is relatively straightforward. You get the highest multiplier from doing quests, and quests give more experience at higher levels. So do the highest level quests you have available.

Once you reach level 50 on a character, the amount of legacy experience you gain from PvP and killing NPCs drops off significantly, but the amount earned from questing stays relatively the same. You should complete all of the solo-able quests on Corellia and Ilum if you haven't already. Also be sure to do your daily quests and space missions - both for Daily Commendations and excellent Legacy XP. If you can, group up and do the daily and weekly dungeon and raid quests as a Level 50 'Very Hard' quest will earn you roughly 6035 legacy XP. If you've completed your dailies and still have time, level an alt via questing.

Changes in Patch 1.1.2
The Public Test Server was recently updated with Patch 1.1.2 and the notes indicated that the required experience for leveling your legacy past level 20 has been increased. Well, the increase is significant. In Patch 1.1.1 each level only required 1000 more legacy experience than the previous level. In 1.1.2 the required experience ramps up considerably starting at Level 20. Rather than a 1k experience increase the required experience will increase either by 16k or 17k. The net result is a roughly 50% increase in total legacy experience required to reach legacy level 50.

Assuming they don't adjust people's legacy levels down in the coming patch if you are above Legacy Level 20 you should consider the time between now and the release of 1.1.2 as giving you a 10-100% legacy experience bonus - so lock in those levels while you can!

Table A. Legacy XP Multipliers

Table B. Legacy Level Experience Requirements