Specials are attacks that consume Power to perform. The deal damage according to the Damage stat, upgrades, hit counter, and other modifiers.
Every hero has a special 1, special 2, and special 3. Special 3 is unlocked when a hero reaches the Star rating of 3 (regardless of tier; most gold heroes start already at 3 stars). The term abilities include Passives, which are effects that are always active and do not cost power to use, as well as Supermoves which are a similar but different class of attacks.
Note the classification of special 1, 2 or 3 depends on their position on the hero's ability menu and their upgrade cost (250, 350, and 500 credits respectively at level 1 to 2), not their power cost - for example, Cyborg's special 2 costs less power than his special 1 (although almost all other hero have the same order for both). Special block drains power constantly while used, and is always the special 2 to any hero who has it.
Specials can be upgraded with credits, but the special's level cannot be raised to above the hero's level (e.g. a level 20 hero can only have level 20 abilities at maximum), up to 70. Passive level is capped at the Star rating of the hero instead (up to 5, cannot be upgraded to 6 even at 6-star), while Supermoves are automatically upgraded according to star rating.
Special 1 costs 1.18M (million) credits to reach level 70. The cost to max special 2 is 1.42M, and 1.79M for special 3. A detailed table can be seen here.