It started out life as St. Peter's Abbey, and between 1042 and 1052, the saintly King Edward the Confessor, ordered the rebuilding of the abbey church into something a bit more, well, fit for a king, especially a king's tomb. The original church was built on what was known as Thorn Ey (Thorney Island) and was, indeed at the time, an eyot, a little island in the Thames. Though it still wasn't finished by the time Edward died, it was consecrated on 28 December 1065, a week before the king passed.
The church we know today was begun nearly two hundred years later by Henry II in 1245. And kept getting updated and remodeled through the fifteen hundreds, though a great deal of what we see today was done by architect Henry Yevele during Crispin's day, the Purbeck marble for the interior columns being commissioned by his old friend, Abbot Nicholas de Litlyngton.
Westminster is a minster, which is a church with a monastery. But which came first, the name of the church or the name of the town that grew up around it? At any rate, Westminster Abbey got another designation in 1540 by Henry VIII giving it cathedral status so it couldn't be dissolved as he dissolved so many monasteries. (Though cathedrals are usually designated so because it is the seat of a bishop, the cathedra and this wasn't the seat of a bishop. Hey, it's good to be king.) And then in his daughter's time, Elizabeth I dedicated it as a "Royal Peculiar" which makes it directly responsible to the sovereign as opposed to a bishop, and headed by a royally appointed dean, making it the "Collegiate Church of St Peter," and remains so today.