Refers to the personal name as opposed to the family name (上の名前). Common in everyday speech when asking or telling someone's given name.
May I call you by your first name?
His given name is Taro.
Compound of 下 (した, 'below') + の (possessive particle) + 名前 (なまえ, 'name'). The expression contrasts with 上の名前 (うえのなまえ, 'family name'), reflecting the traditional Japanese name order where the family name comes first (above) and the given name comes second (below).