The country is England. Its language is called English. Not British. Great Britain contains the countries of Wales and Scotland too, which have their own languages but use English for convenience, so there is no British English. Received Pronunciation (RP) is the only British accent that has no specific geographical correlate: it is not possible, on hearing someone speak RP, to know which part of the United Kingdom he or she comes from. There are variations of English, like regional dialects. American is a variation of English. The dialect regions of the United States are most clearly marked along the Atlantic littoral, where the earlier settlements were made. Three dialects can be defined: Northern, Midland, and Southern. Each has its subdialects.
As such, American English is just a variation of English, the same as in Canada, Australia and so on.