Dining is the act of eating a meal, particularly in the evening. If you want to impress someone with your formality, ask them if they'll do you the honor of dining with you.
The noun dining is from dine, "to eat the last meal of the day." The Old French source, disner, originally meant "take the first meal of the day," from a Latin root meaning "break one's fast." These days, most people think of dining as an evening activity, one that generally happens at a dining table in a dining room!