Every character has a special 1, special 2, and special 3. The term abilities include Passives, which are effects that are always active and do not cost power to use. Supermoves are a similar but different class of attacks.
Note the classification of special 1, 2 or 3 depends on their position on the character'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 characters have the same order for both). Special 1 and 2 are always available, while special 3 requires the character to reach 3 stars.
Specials can be upgraded with credits, but the special's level cannot be raised to above the character's level (e.g. a level 20 character can only have level 20 abilities at maximum), up to 70. Passive level is capped at the Star rating of the character instead (up to 5, cannot be upgraded to 6 even at 6-star), while Supermoves are automatically upgraded according to star rating.