A person's "ID" can mean their identification card. In the United States, a driver's license is the most commonly used ID because it shows a person's name along with a photo, and most people have one.
You might "check" someone's ID to make sure that they are the person they claim to be. Another common reason to check someone's ID is to look at their birth date to make sure that they're old enough to buy alcohol:
Yesterday I was buying a case of beer and the guy at the checkout asked to check my ID. I was like, "don't you see this gray hair?"