To vaccinate is to immunize someone against a disease. Babies are usually vaccinated against many diseases soon after birth.
A vaccine is a type of medicine that makes you resistant (immune) to a disease like polio. To give someone such a medicine is to vaccinate them. You can use this word in a few ways. A doctor is usually the person who vaccinates a child (or older person) by administering a vaccine. You can also say the parents have vaccinated their children by making the decision to do so. Animals can be vaccinated too, like dogs are vaccinated against rabies.