To asphyxiate is to keep someone from breathing — to suffocate or smother them. Choking on a piece of food can asphyxiate a person.
Your body needs to breathe oxygen in order to function, and when it's deprived of oxygen that's called asphyxia or asphyxiation. If a person stops being able to breathe, you can also say they asphyxiate. The origin of this verb is asphyxia, "stopping of the pulse," in Greek, which originally had the same meaning in English, before it came to mean "suffocation."