Someone who's abstinent is moderate and restrained, especially when it comes to consuming food and drink. A person who quits drinking alcohol completely is abstinent.
It's most common to use the adjective abstinent to describe someone who has given up using alcohol or other intoxicants, but you can also use it to talk about a person who's self-restrained in other ways. You might say, "No dessert for me — I'm trying to be abstinent this month!" Abstinent is an Old French word meaning "moderate" or "modest," from the Latin abstinere, "to refrain from."