Toothpaste is a product that is used to clean the teeth as well as the gums and can also be used on the tongue. Some earlier teeth cleaning solutions from historical times would not be palatable today. Toothpaste is usually applied with a toothbrush.
It is designed to help remove plaque and bacteria, and to counteract the acid which that bacteria has produced. Some toothpastes also try to whiten teeth, kill the bacteria, and/or expose the teeth to fluoride. Fluoride will try to bond with the calcium in the teeth, thus theoretically making them harder. However, some say that this actually weakens the teeth, perhaps because the fluoride loosens the calcium's bond to the rest of the tooth. In any case, fluoride is in plentiful supply, as it is an artificial byproduct of the refining of Aluminum so it remains a common, cheap substance in many things.