Engaged is an adjective. It literally means to be occupied and in other sense it means to agree to get married. It is the past form of the word engage. The synonym for engaged are busy, reserved, booked, etc.
Engaged gives the strong sense of being busy. The term also depicts the relationship status of a person. Engaged means a couple has promised to get married after the ritual of engagement. It shows the bound of two people soon to be sharing the marital vows. The couple after being engaged are known as Fiance.