Black English
Black English is a very broad term used to refer to British and American English as spoken by the black communites in the US and the UK.
Interest in the existence of Black English began in the early twentieth century with several publications by linguist George Philip Krapp.
Krapp attributed the existence of Black English to the "baby-talk" that he assumed slave masters must have employed when speaking to their slaves. He hypothesized that slave masters addressed their servants in a simplified English, similar to that used with babies.
The view that black people who spoke in Black English were deprived of a real, dynamic, and multi-faceted language continued through the 1960s untill William Labov, a linguist at the University of Pennsylvania, followed by J.L. Dillard, Geneva Smitherman, and other linguists, mounted an impressive defense of the legitimacy of Black English refuting the language deprivation theories of previous times.
US & AAVE
African American Vernacular English (AAVE) is a distinct variety of English rather than a dialect. The variety is also known as "ebonics" though this is not a term used by linguists.
AAVE has its roots in the slave trade where people captured in various parts of Africa and with a variety of languages were forced to create a pidgin or creole - a common language composed of fragments of their native languages. Eventually this incorporated elements of English so it could also be used to communicate with the slave owners.
UK & BBE
British Black English (BBE) has some similar origins but is based on a Jamaican creole spoken by Caribbean communities, mainly in London but also in large cities such as Manchester, Bristol, Birmingham, Leeds and Nottingham.
There is a history of British sugar planters in Jamaica, which was a British colony until 1948, after which in the 1950's there was significant immigration to London as England welcomed workers in its post war expansion and rebuilding. Jamaican Creole is recognised as an independent variety with its own grammar-system and vocabulary.
Increasingly British Black English speakers are finding their own voice in literature such as rap poetry and song, with Benjamin Zephaniah a respected name. There is no standard form of orthography so much of the language is written semi-phonetically - "yuhself" for "yourself" "dat" and "dem" for "that" and "them", "nuff" for "enough", "respek" for "respect".
Some of their culture appeals to young native white English speakers who in turn adopt features of BBE speech mixed with their native Cockney.