Differences between British and U.S. usage: In English law, a brief is a document by which a solicitor instructs a barrister to represent a client in court.

There is a definition in the Oxford Dictionary of Law, edited by Elizabeth A. Martin, ISBN 0 19 860399 1, a paperback that has not much increased in size since it used to be called the Concise Oxford Dictionary of Law. Here’s the complete quote via xrefer:

brief
A document by which a solicitor instructs a barrister to appear as an advocate in court. Unless the client is receiving legal aid, the brief must be marked with a fee that is paid to counsel whether he is successful or not. A brief usually comprises a backsheet, typed on large brief-size paper giving the title of the case and including the solicitor’s instructions, which is wrapped around the other papers relevant to the case. The whole bundle is tied up with red tape in the case of a private client and white tape if the brief is from the Crown.
Dictionary of Law, Oxford University Press © Market House Books Ltd 1997

(So the Oxford Dictionary of Law can be consulted online, together with a number of other sources).

‘Brief-size paper’ is rather a circular definition. It used to be paper twice as wide as the normal foolscap, typed on on a huge typewriter. We used to call the red tape ‘pink tape’ (see entry on stationery). A vague idea can be obtained from the picture at the mysterious Virtual Chambers site.There are other one-volume dictionaries of English law, also quite good, often with more but shorter entries than the Oxford.

In U.S. usage, a brief is usually:

a formal written presentation of an argument that sets forth the main points with supporting precedents and evidence. Briefs are filed either by a party or an amicus curiae with a court usu. regarding a specific motion… or point of law.

From Merriam Webster’s Dictionary of Law, 1996, ISBN 0 87779 604 1
(It also gives the meaning: a concise statement of a client’s case written for the instruction of an attorney usu. by a law clerk, which is like the English meaning, but I’ve never encountered this meaning).

Here’s an online definition from Nolo Press:

A document used to submit a legal contention or argument to a court. A brief typically sets out the facts of the case and a party’s argument as to why she should prevail. These arguments must be supported by legal authority and precedent, such as statutes, regulations and previous court decisions. Although it is usually possible to submit a brief to a trial court (called a trial brief), briefs are most commonly used as a central part of the appeal process (an appellate brief). But don’t be fooled by the name — briefs are usually anything but brief, as pointed out by writer Franz Kafka, who defined a lawyer as “a person who writes a 10,000 word decision and calls it a brief.”

(The Kafka quote is a bit suspicious – what German word would he have used?)

This would be some kind of pleading in British English (although the word pleading, like plaintiff, was declared out of date by the new Civil Procedure Rules in 1999.