Origin: Danish, Armenian, Persian
Meaning: a Danish short form of Katherine, which has several possible meanings: “pure, chaste” from Greek katharos; or it could come from Greek hekateros “each of the two; singly”; or from Greek aikia meaning “torture”. It could also be a Coptic name meaning “my consecration of your name”.
Karen is also a masculine name in Armenian, a transcription of Garen, itself a short form of Garegin, of unknown meaning. It's also derived from a Persian masculine name possibly meaning "brave" or "valorous" in Persian. Karen could also be derived from an ancient Armenian masculine name, Kar, possibly meaning "stone".
Karen also seems to be a popular name in parts of Africa and Asia, the former likely a variant spelling of Karin, a Japanese feminine name with various meanings depending on the kanji used; it's also a homonym for a Japanese quince, the name of a fruit.
It could also be a variant spelling of Keren, a Hebrew feminine name meaning "horn" or "ray of light".