The English/British court was always peripatetic (and arguably still is today, as the present Queen divides her time between Buckingham Palace, Windsor Castle, Sandringham, and Balmoral). There were numerous royal residences, some of which are no longer standing, and different palaces came in and out of favour depending on the monarch. Pre-Stuart monarchs also spent considerable periods of time on progress, travelling the country with their suite and staying in the houses of favoured courtiers, as a way of showing themselves to the people.

Speaking very broadly, Windsor Castle is probably the most significant royal residence, in that it has been used most extensively and consistently by successive monarchs. Although it was first built by William the Conqueror, it wasn't used as a royal residence until the reign of Henry I, and was extended and refurbished by Henry II in the later 12th century. Over the next 800 years, it was gradually extended and updated, with St George's Chapel (the burial place for British royalty) begun by Edward IV in the 15th century. It was substantially remodelled in the 'Gothic' style in the late 18th century, by George III.

A series of smaller palaces was used in the later medieval period, among which Eltham Palace and Placentia, at Greenwich, were probably the most important. Placentia, which was demolished in the 17th century, was the birthplace of Henry VIII. Eltham is still standing.

Henry VIII was responsible for acquiring the two palaces which would become the most significant residences of the later Tudor and Stuart monarchs. By the time of his reign (1509-47), the principal London residence of the monarch was the Palace of Westminster, where Parliament also met (and still meets, as the 19th century Houses of Parliament stands on the site now). The residential part of Westminster was destroyed by fire in 1512, and Henry later 'acquired' the neighbouring house of York Place from Cardinal Wolsey, after the latter was toppled from power in 1530. This was expanded and became the palace of Whitehall, which was a vast, rambling complex of buildings. Whitehall was the principal London residence of the monarch until the 1690s, when a series of fires reduced it to ruins. The only surviving part was the Banqueting House, built by Inigo Jones in 1619-22 for James I/VI, and the first 'classical' building to be constructed in England.

The other palace acquired by Henry VIII was Hampton Court, again 'acquired' (seized) from Wolsey as he fell from power. Hampton Court is about 15 miles outside central London, far enough to be rural but close enough to be at hand in the event of a crisis. It was extended substantially in the 1680s and 90s by William III and Mary II, the extensions designed in the Baroque style by Sir Christopher Wren (who also rebuilt St Paul's Cathedral). George II (1727-60) was the last monarch to live at Hampton Court for any significant length of time.

After Whitehall Palace burned down completely in 1698, the official London residence became St James's Palace, which had previously been a minor residence, built as a retreat by Henry VIII. St James's remains the 'senior' royal palace today, and foreign ambassadors are technically ambassadors 'to the Court of St James's', but it was rarely used as an actual residence. By the time George III came to the throne in 1760, he purchased Buckingham House (as it was then), as a private urban residence for his family, with official business conducted at St James's. He also remodelled Kew Palace as a rural retreat.

That was the point at which Buckingham Palace grew into the best-known royal palace, because of its London location and public visibility. It was extended (the royal family love extensions) by George IV, at which point it became a 'palace' rather than a house, and Queen Victoria was the first monarch to live there. The famous facade with the balcony wasn't actually built until the second half of the 19th century.

In addition to Buckingham Palace and Windsor, Queen Victoria also acquired Balmoral House, in Scotland, and Sandringham, in Norfolk, which subsequent monarchs have also used as their country retreats. After the early death of Prince Albert, she spent a lot of time at Osborne House on the Isle of Wight, but this was sold off and is no longer a royal residence.

The pattern set by Victoria has more or less persisted to the present monarch. As you can see, it's a long and complex answer, and doesn't even begin to address some of the smaller/less important palaces, like Richmond; or Holyrood, which is the official royal residence in Scotland, although little used as an actual residence.