Abilities Are like extra passives that your Pokémon can have. Each ability impacts the way that your pokémon performs in battle.
Most Pokémon only have 2 Abilities, while some can have 4 Abilities and a Secret Technique.
All the pokémon start with one ability, and the Second Ability is unlockable when you Unlock Master Dungeons. These Master Dungeons are the ones who will provide the items necessary to unlock the Second Ability:
Once you meet the requirements, the Second Ability can be unlocked.
In order to change your pokémon Abilities, you either have Ability Cards that can drop from multiple Adventure Maps, Master Maps and also Fated Enemy, or, you can spend diamonds directly to change Abilities.
Master Ability Cards which are the ones with the best rarity rate, are also very rare to drop from the Maps. So players usually prefer to either wait to obtain those in Events, or to spend diamonds directly.
About Third and Dream Abilities, those can also be changed by spending diamonds, but their Ability Cards are obtained in a different way: playing on the last 2 islands of Cross Server Team. Those islands will provide materials to unlock and improve these 2 abilities.
To learn more about Dream Ability, Third Ability and Secret Technique, check the video bellow: