'A brief' (in this context - design) is a specification of work to be done. There can be less specific kinds of brief (e.g. His job was not to involve himself in the day-to-day business; he had a watching brief.).
In a legal context, brief has a particular meaning, referring to the lawyer's knowledge of a case (those parts of his client's affairs that pertain to the case, excluding other matters: I didn't know about his tax affairs - they weren't part of the brief. I believe this is where the word briefcase comes from - but don't quote me, as I haven't looked it up (yet)