The private rooms of the king or lord of a *burh, where he could be private and away from the hall, which was always public. Sometimes it was a small building physically separate from the hall, containing private rooms. More poetically, it was later a kind of summer-house wrapped in flowers and foliage for the ladies of the court.
Cf. Burthegn
Dictionary of Medieval Terms and Phrases. Christopher Coredon with Ann Williams.