Well there's moves which are abilities which are learnt
Abilties however is something different.
Each Pokémon has an ability which is like a "passive" which differs between each Pokémon.
Abra has +5 ATTACK and Alakazam deals % increased damage with psychic type moves for an example.
By breeding no. I haven't assigned genders to the Pokémons. If I had assigned genders to the Pokémons and some other stuff.
Pokémon eggs are items you can buy from a store in the 4th city.
They spawn a random pokémon ex. A level 1 Snorlax.
The advantage is that this Pokémon has not had any of the stats you gain per level assigned to it yet allowing you to train your Pokémon how you want it in whatever stats you want to.