Received Pronunciation - commonly called RP or Standard British - is the accent most people mean when they say that they want to learn a British accent. RP can be heard throughout England and Wales, though it's primarily an accent of southern England. It's commonly considered the Standard English accent, though it's still considered a non-regional dialect of England.
Originally, it was likely the accent of the East Midlands region of England (see the photo to the right) back in the 15th century. It spread further than other accents of that time because of the prosperous trade of that region, sending the accent further afield and associating it with economic success. As trade became more and more centered around London and spread towards the south of England, the accent found its new home there.
Teaching Received Pronunciation
Though this Standard British accent has long been the accent of money and influence, it became especially associated with the power of England in the 20th century. This is largely due to a major shift in education in the late 1890s.
At that time, there was a movement to standardize the accent and dialect of students in the educational system of England. A committee was assigned to establish the standardized British accent, and RP narrowly won the vote - the Yorkshire accent just missed the honor.
The accent was also adopted as the BBC standard accent in all broadcasts. It was considered the most universally understood accent in England, and also became considered the most pleasant accent to listen to. It remained the primary broadcasting accent until the 1980s, when there was a movement towards allowing more regionalisms on the air.
Received Pronunciation by any other name...
The term Received Pronunciation was popularized by Daniel Jones in his 1926 edition of the English Pronouncing Dictionary. Rather than meaning given, Received Pronunciation refers to the fact that it is the approved or standard pronunciation. (This is an older definition of the term "received" which is no longer a standard usage.)
There are a number of terms that can describe variations in Received Pronunciation. The Queen's English is one term for the royals' mode of speech, which is definitely a certain strain of RP - the current Queen has some atypical pronunciations in her speech which make it fall outside of what might traditionally be considered Received Pronunciation.
You can hear her pronunciation of "around" in this video at :28 back in 1957 (also listen to "how" at :41). You can hear her say "round" again at 1:38 in 2007. This isn't necessarily an element of the RP accent, but is rather a part of the Queen's idiolect.
Variation within the accent is common: Some consider Cambridge and Oxford English to be different accents, as learned at the two universities, and Sloane Rangers, or Sloanes, are usually centered around north London (named for Sloane Square in Chelsea). Are they truly different accents? It all depends on how narrowly you slice the accent pie... In the end, there's quite a range of speech that actually fits into this broad target of Received Pronunciation, no matter what you call it.