Johann Nicolaus Forkel, Sebastian’s first biographer, wrote about the “six great suites” that they “are known by the name of English Suites because the composer composed them for an Englishman of rank. They have all great value as works of art, but some single pieces among them, for example, the gigues of the Fifth and Sixth Suites, are to be considered as perfect masterpieces of original harmony and melody.”
Forkel’s idea of a commission by a prominent Englishman is probably faulty supposition, however. It may have been that these suites were inspired by Bach’s study of the six harpsichord suites by Charles Dieupart, a French composer who worked in London at the time. Bach copied out those six suites sometime between 1709 and 1714, and he used one of Dieupart’s Gigues as the model for the Prelude of the first English Suite.
Here is pianist Andras Schiff performing the English Suites by Bach: