From Destinypedia, the Destiny wiki
Abilities are the various powers or weapons that beings across the Destiny universe are able to use.
Abilities are classified as techniques, attacks, weaponry, or powers that are not the primary handheld weapon the entity using them attacks with. This includes grenades, movement abilities, unique attacks/gimmicks used by Raid and Strike bosses, and more. The use of abilities can be stopped with Suppression effects, and their regeneration slowed by Slow effects, Modifiers or Anti-Overload weapons.
Guardians can choose their abilities based on the Subclass they use. All Guardian abilities deal Elemental damage, and have a cooldown period. The effects of Guardian abilites can be modified, even drastically, by various effects, namely Aspects, Fragments and Exotic Armor.
Movement abilities define how a Guardian jumps. Hunters can do multiple jumps across the air, Warlocks glide in the direction they are facing, and Titans move upwards in a burst. Movement abilities are shared across the same class. All subclasses share three, but can have their own exclusive ones.
Grenades are explosive abilities that deal heavy or continuous damage, weaken the enemies, or stenghten their allies. They are shared across classes. Their cooldown depends on Discipline.
Melees are abilities connected to close combat. Some melee abilties create visible effects, while others simply increase the power or utility of a basic melee attack. They are exclusive to a class' subclass. Their cooldown depends on Strengh.
Introduced in Destiny 2, Class Abilities are moves that define a class. All subclasses share two, but can have their own exclusive ones, with Hunters are capable of quick Dodges, Warlocks can create Wells to increase their damage or regeneration, and Titans can create temporary Barricades. Their cooldown depends on the Class' stat (Mobility, Resilience or Recovery).
Super abilities are game-changing moves that enable Guardians to survive overwhelming odds. Supers allow Guardians to deal massive damage in an instant, kill huge hordes of enemies, or enormously increase their strength and resilience. All subclasses have at least one super. Super cooldown depends on Intellect.
Some subclasses allow the Guardian to use an additional move while in the air. Examples are the quick Icarus Dash or the downward Shatterdive. They usually have fixed, short cooldowns.
Enemy abilites are defined as all the attacks enemies do without using their wielded weapon or fists, as well as most special movement modes like the Vex's teleportation. This includes the use of Grenades. Enemy abilities have cooldowns as well, though less apparent than a Guardian's. While enemy abilities are exclusive to one base enemy type, Ultra enemies can often use abilites of other enemies, such as how Crota, Son of Oryx uses a Wizards Darkness Blast, and they often possess abilities crucial to the progression of their battle.