From Ancestry.com:
English, Scottish, and Irish: generally a nickname referring to the color of the hair or complexion, Middle English br(o)un, from Old English brun or Old French brun. This word is occasionally found in Old English and Old Norse as a personal name or byname. Brun- was also a Germanic name-forming element. Some instances of Old English Brun as a personal name may therefore be short forms of compound names such as Brungar, Brunwine, etc. As a Scottish and Irish name, it sometimes represents a translation of Gaelic Donn. As an American family name, it has absorbed numerous surnames from other languages with the same meaning.