The grammar structure たまえ is a mild command form used by a speaker of higher status (such as a boss, teacher, or father figure) to direct someone of lower status to do something. It attaches to the verb stem (ます-stem) and carries a somewhat authoritative yet not harsh tone – softer than the bare imperative (しろ / やれ) but still clearly a command from above.
たまえ is most commonly encountered in fiction, dramas, and older or formal speech. It evokes the image of a distinguished or paternal authority figure – a professor addressing students, a company president addressing employees, or a father addressing his son. In modern everyday conversation, it sounds old-fashioned or literary, and using it casually can come across as arrogant or humorous.
The pattern is simple: take the ます-stem of a verb and add たまえ.
Because of its literary and somewhat archaic flavor, たまえ is most likely to appear in written Japanese, period dramas, anime/manga featuring authoritative characters, and formal speeches.