The Legacy System explained
World of Warcraft: Forever adds the Legacy System: account-wide progression that rewards exploring everything the game has to offer, instead of only chasing more power. Leveling alts helps, but you can unlock everything on a single character too.
You unlock the system with your first Legacy Point, for most players when their first character reaches level 25. Open it with the shield icon in the menu or press Y.
Legacy Challenges
There are 65 challenges in six categories. Each one gives 1 Legacy Point, shared by all your non-Hardcore characters.

- Classes (27): reach level 25, 45 and 60 with each class.
- Tradeskills (18): reach 150, 225 and 300 in each non-gathering primary profession.
- PvP (12): reach PvP ranks 3, 7, 10, 13 and 14, Exalted with each battleground reputation, and weeks 4, 7 and 10 of the Field of Honor season.
- Adventure (2): explore the whole world map on one character, and finish the Lord Valthalak dungeon set questline.
- Dungeons (3): beat the last boss of every dungeon in the level 15 to 25, 26 to 45 and 46 to 60 ranges.
- Raids (3): defeat Onyxia, all 13 bosses of Hyjal Summit and all 8 bosses of Barrow Deeps.
That last one is where Legends of Azeroth comes in.
Legacy Perks
Each character spends the account's points on its own perks, in three trees. At launch you can spend up to 16 points per character, so new players do not fall too far behind and you never need alts to max out. One character alone can earn up to 29 points.

Professions: more skill-ups (also Lockpicking), extra rare gathering finds, cheaper vendor items, more Merchant's Favor from the new trade factions (for the Horde: Durotar Supply and Logistics), extra cooking results, extra fish and a daily cooldown to level tradeskills faster.
Adventure: a bigger and faster rested XP pool, a bit of health and mana back after each kill, better stealth detection, a shorter cooldown on camping features, shorter Recently Bandaged, talents up to 5 levels early and cheaper, faster flights.
Resourcefulness: longer food buffs, faster corpse runs with free buffs after a wipe, less durability loss, more Honor, longer party, raid and camp buffs, more reputation, and Reagent Economy: no more purchasable reagents for class spells.
Legacy Rewards
Your total points also unlock cosmetics on a reward track. At launch:

- Replica Ironforge Air Rifle, a toy for shootouts with other players.
- Spectral Bear Cub, a pet.
- Spectral Bear Tabard.
- Reins of the Spectral Bear, a mount.
Hardcore
Hardcore comes after launch with its own rules. Challenges from other rulesets do not count there, but challenges earned in Hardcore count everywhere. Since PvP challenges are impossible in Hardcore, it gets 12 challenges of its own.
More to come
Every content update will bring new challenges, new perks and more rewards, and later even new perk trees.
Source: Blizzard's full article at https://worldofwarcraft.blizzard.com/en-us/news/24307383/get-to-know-the-world-of-warcraft-forever-legacy-system
Images: Blizzard Entertainment.