Abilities are unique powers you learn separate from your weapons and fruits: air jump, 
Aura (Haki), 
Instinct... They give permanent advantages both in combat and exploration. Knowing where to buy them and in what order is the foundation of proper progression.
How to get it
The first three are bought from the 
Ability Teacher, in a cave in 
Frozen Village or 
Magma Village: 
Air Jump (10,000 Beli), Aura (25,000 Beli), and 
Flash Step (100,000 Beli).
How It Works
Beyond the basic trio, Instinct is bought for 750,000 Beli from the 
Instinct Teacher, in the grand temple at the bottom of the second sub-island of 
Upper Skylands: it grants vision of enemies through walls and evasion of certain attacks, with training to unlock its full potential. Its Instinct V2 version unlocks by completing the 
Hungry Man's Quest (
Third Sea only, 5,000,000 Beli) and notably allows you to see other players' hotbars.
Each race also has its own ability, unlocked at V3 stage via NPC Arowe: 
Human = 
Last Resort, 
Rabbit = 
Agility, 
Angel = 
Heavenly Blood, 
Shark = 
Water Body, 
Ghoul = 
Heightened Senses, 
Cyborg = 
Energy Core. 
Draco unlocks 
Primordial Reign via the 
Dragon Wizard, not 
Arowe.
Finally, 
Summon Sea Beast unlocks at 10 million 
Bounty/Honor: it summons a 
Sea Beast at will (with cooldown), useful for PvP at sea, available in Second and Third Sea.
Good to Know
There are also 
Admin Abilities, reserved for the game's staff: you'll hardly ever encounter them. For details on each ability (Aura, Instinct, Air Jump...), find the dedicated pages in the mechanics section of the wiki.
Abilities
Accessories
Admin Panel
Admin Terminal
Allying
Area of Effect
Auto Click Ability of Weapons
Awakened Bosses
Baits
Ban Restore
Beast
Blox Fruits
Blox Fruits "Stock"
Blue Gear
Blue Moon
Boats
Books
Bosses
Brazil World
Chess