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. 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). Passive level is capped at the star level of the character instead, while Supermoves cannot be upgraded at all.