"Mi amore" is an Italian expression that, in English, means "my love." Though it is typically an expression of love, it is sometimes also used to express extreme affection.
Amor is the Latin verb for love. French, Italian, French, Portuguese and Spanish are all Romance languages, meaning that they have a Latin base. All of these languages use a version of amore to express love, though in Spanish "te Quiero," which actually means "I want you" is the more prevalent casual expression. In French, "ami/amie," the word for "friend," is also very similar to amore, again playing on the close relationship between love and strong affection.