In AtLA, bending is a synchronisation between body, mind, spirit, and the world itself. It's based on various ancient Eastern philosophies, religions, and pseudo sciences. One's connection with the world is expressed through various forms of martial arts meant to direct and compel an element by manipulating the connection between the bender and their patron element. It's notable that if body, mind, or spirit is ever not in synch with the bending, it will fail.
The connection between body and element is one passed down through bloodlines, though not all children inherit the skill. More than that, a bloodline can have a stronger or weaker connection meaning that stronger benders will have stronger bender children. Again, this is based on Eastern ideas that placed a great deal of importance on lineage as a spiritually relevant factor.
In Korra it's magic, and all children get magic from their parents.