Abilities are what make cards stand out from the mundane, to be unique and interesting rather than dull and bland. Whether being an offensive attack ability or an augmentive special ability, they define what the cards are and what they are capable of doing.
All abilities come in four types, or "modes" which dictate how it behaves and is operated.
Passive Ability
Passive abilities activate and persist automatically, without needing interference from a player. Sometimes these passives only come into effect when another ability is used.
Auto-Cast Ability
Auto-Cast abilities are activated automatically when a target of suitable nature is in range. These abilities usually have a cooldown and are usually ranged attacks, with some exceptions.
The 
Kobold Engineer for example: Can only use its architect ability on friendly structures, rather than enemy creatures.
Activatable Ability
Activatable abilities require manual activation to take effect, rather than being performed automatically. They are one of the most common ability types, present on most creature cards. These abilities often cost power to activate, requiring strategic use for power efficiency.
Toggled Ability
Toggled Abilities are similar to Activatable abilities, but they instead change the units mode which affects its behavior and may activate other abilities. A prime example are creatures with the Root ability, which immobilizes them but changes their attack to a much stronger but slower form.