"Dink" is a word that stands for "double income, no kids." It means that two grown-ups who are married or in a relationship are earning money, but they don't have any children. That's what "no kids" means. "Double income" means that both of them are working and earning money.
Sometimes people use this word to describe a certain lifestyle that some couples might have. They might have more money to spend on themselves because they don't have to spend it on kids. They might be able to go on fancy vacations or buy nice things. But just because they don't have kids doesn't mean their lives are easier or better than people who do have kids. It's just a different way of living.