A rice paddyfield. You'd think this would refer only to a rice field, but it actually refers to other types of fields too, making this word simply field. I guess that shows how important rice is, because if a field isn't growing rice, it ain't no field o' mine.
The readings come from what you learned with the kanji. 畑 uses the shorter はた reading that you learned.
Nowadays ploughing fields without any machinery is a waste of time.