The Court of St. James's is named after St James's Palace which is the senior Palace of the Sovereign, currently Queen Elizabeth II. It remains the official residence of the British Monarchy despite Queen Victoria moving to Buckingham Palace after her accession in 1837.
Though St James's Palace is the official residence of the Sovereign, the Court moves with The Queen. As Buckingham Palace is the official London residence, the court and consequently meetings of the Privy Council occur there as the Queen does not live at St James's. During her extended stays at Windsor Castle (typically during Easter), Sandringham during Christmas and at Holyrood Palace or Balmoral Castle in Scotland in the summer the Court will be at Windsor, Sandringham, Holyrood or Balmoral. When the Queen travels overseas, the court also travels with her.
All Ambassadors and High Commissioners to or from the United Kingdom are accredited and received to the Court of St. James's and as it would be infeasible to do this to a moving Court so a static name was needed. The official residence was therefore chosen as the Marshal of the Diplomatic Corps (before 1920, Master of the Ceremonies), who acts as the link between the Queen and the foreign diplomatic missions, is permanently based there.
There are currently 153 foreign missions, manned by 2,382 diplomats, accredited to the Court of St James's in London. This total is made up of 46 high commissions (missions from Commonwealth countries) and 108 embassies (missions from non-Commonwealth, foreign countries).
The court is often incorrectly described as the "Court of St. James."