Refers specifically to a son of the Japanese emperor or a male member of the imperial family. Not used for princes of foreign monarchies, where 王子 (おうじ) is more common.
He is the emperor's second son.
I hear that the imperial prince will visit abroad next month.
Last month, an imperial prince was born into the royal family.
皇子 refers to a son of the Japanese emperor, the male counterpart to 皇女, a daughter of the emperor.
皇子 and 王子 are both pronounced おうじ, but 皇子 refers only to a son of the Japanese emperor while 王子 is the general word for a prince, including foreign or fictional ones.
皇子 simply means a prince (an emperor's son), while 皇太子 specifically means the prince designated as heir apparent.
counter for people
Compound of 皇 (imperial) and 子 (child). The reading おうじ is the on'yomi of both characters.