To guarantee is to promise or to a make binding agreement. If a baseball player guarantees his team will win the World Series, he better not lead the league in strikeouts.
Guarantee is a word that is both a noun and a verb: the noun means "a binding agreement" and the verb is the act of making that agreement. This word seems to have come from garante, the Spanish word for "protect" and it's related to the Old French word warrant, which means "to authorize." Today, we often come across guarantee in advertisements, and you might wonder how they might guarantee to make your social life better if your teeth are whiter!