May I invite you to dinner?

Literal

Dinner [to-に] invite-[permission-てもいい]-be-[polite-ですか]?

A permission-seeking invitation. 誘う ('to invite, ask out') is the neutral verb for inviting someone — more casual than 招待する. Combining it with ~てもいいですか gives an extra layer of politeness: 'would it be okay if I asked?' A graceful way to ask in social situations.