What does it mean to be on the battlefield and in your hand?

The creature exists in both zones. It's considered on the battlefield for anything that cares about permanents being on the battlefield and it's considered in your hand for things that care about cards in your hand.

Does Masterful Ninja's ability cause any enters-the-battlefield abilities to trigger? What about leaves-the-battlefield abilities when the turn ends and it goes back to being just in my hand?

Masterful Ninja is so masterful that any abilities that care about a creature entering or leaving the battlefield don't even notice it. (If Masterful Ninja enters or leaves the battlefield another way, those abilities work as normal.)

What happens if I try to move Masterful Ninja from battlefield to hand or vice versa?

Masterful Ninja stays in both zones. For example, say your opponent casts Unsummon, trying to return Masterful Ninja from the battlefield to your hand. It's already in your hand, so nothing happens. Masterful Ninja stays on the battlefield. And in your hand.

What happens if Masterful Ninja is destroyed?

Masterful Ninja goes to the graveyard and will no longer be in your hand or on the battlefield.

What happens if Masterful Ninja is discarded?

Same thing. It'll be in just the graveyard.