Despite not being officially recognised, Pidgin is widely spoken and understood in West Africa; in Nigeria it is estimated that between 3 and 5 million people use it as their first language when carrying out day-to-day interactions, and it is the second language for many millions more. It's also spoken in Cameroon, Ghana and parts of Sierra Leone. Peter Okwoche, anchor on the radio programme Focus on Africa, was on the Today programme this morning (listen here, he was on at about 07:45, i.e. 1hr 45 mins into the programme).
Okwoche said that the language developed when traders first landed in West Africa, in order to allow them to interact with the local people. It is made up of bits of English, French, Portuguese, and dialects of West Africa. See this BBC web page for some examples, including, I wan chop (I want to eat), Wetin dey 'appen? (what's happening?) and How body? (how are you?). There is no standardised form of Pidgin, but the BBC will be broadcasting in a variety that everyone understands.
The word pidgin developed from the Chinese pronunciation of business, according to numerous 19th-century sources. Here are three citations from the OED:
1845 J. R. Peters Misc. Remarks upon Chinese vii. 73 Pidgeon, is the common Chinese pronunciation of business.
1850 J. Berncastle Voy. China II. 65 The Chinese not being able to pronounce the word ‘business’, called it ‘bigeon’, which has degenerated into ‘pigeon’, so that this word is in constant use.
1873 Macmillan's Mag. Nov. 45 The strange jargon known as ‘Pigeon English’..derives its name from a series of changes in the word Business... The Chinaman contracted it to Busin, and then through the change of Pishin to Pigeon.
The OED says that the final syllable of business may have been assumed to be a plural inflection, and hence was dropped by the Chinese.
For more, see the BBC page.