たまえ is an order from above: a man of higher standing telling someone below him to do something. It goes on the ます-stem – かき書きたまえ, たべた食べたまえ, したまえ, きた来たまえ – and sounds firm without being rough.
たまえ is old-fashioned. You meet it in novels, period dramas, anime and manga, in the mouth of a professor, a company president or a stern father, often together with きみ君. Said in real conversation today, it comes across as pompous, or as a joke.