The BBC exists in an unusual and subtle relationship with the UK government. It operates under a Royal Charter
http://en.wikipedia.org/wiki/BBC_Charter at one remove from government. It is funded primarily by the licence fee, payable by every household that uses a TV to watch off-air broadcasts.
http://www.tvlicensing.co.uk/check-if-you-need-one/ In the age of multichannel TV the licence has become increasingly controversial but nobody has yet devised a better method. The B&W licence is an anachronism and I suspect it will be abolished when digital switchover is complete in 2012.
The BBC frequently criticises the government and in turn the government criticicises the BBC about bias. Governments of all political hues usually want to bash the BBC but the British people usually trust the BBC rather more than their government.
BBC World and other non-UK services are funded either by advertising or by a direct grant from the government. The government specifies the extent of services (eg it can say "we want a service in Arabic") but the BBC has editorial independence. I suspect there is sometimes considerable tension between the BBC and government when there is news that the government doesn't like. The BBC won its reputation during WW2 by broadcasting bad news as well as good (this is a gross simplification) and has no wish to lose it, along with its global audience, by becoming a simple mouthpiece for the government.