The British royal court has announced that the Duke and Duchess of Cambridge (Prince William and his wife Catherine) have named their son, who was born two days ago, George Alexander Louis. He is a Royal Highness and a Prince of the United Kingdom of Great Britain and Northern Ireland, and will thus be styled as Prince George of Cambridge within Britain (the custom is that princes and princesses who are not the children of the sovereign are known by the territorial designation of their father's peerage).
If Britain remains a monarchy (and Prince Charles does indeed reign as Charles III) the child will one day become King George VII. The first British king of that name was the founder of the current royal house, Elector Georg of Hanover, who inherited the British throne in 1714 and reigned as King George I until his death in 1727. He was succeeded by his son, George II, who outlived his eldest son and was therefore succeeded in 1760 by his grandson, George III, who was again succeeded in 1820 by his eldest son, George IV. Thus the King of Britain was named George consecutively from 1714 to 1830. These four "Hanoverian" Georges had a rather bad press (for instance, George III is remembered primarily for going mad and losing America), although their reputations have been somewhat revised in recent decades.
The name appeared again when Edward VII died in 1910 and was succeeded by his second son, George V (who had not been born to be king). George V's death in 1936 was followed by the brief reign and scandalous abdication of his eldest son, Edward VIII, who upon his abdication was succeeded by his younger brother, Prince Albert, Duke of York. As a mark of continuity and carrying on the traditions of his father after the upheaval of the abdication, Prince Albert chose to be known as George VI, George being the last of his four names.
It might be argued that it was George V who ushered in the current style of monarchy, which has been continued by George VI and Elizabeth II. It is sometimes said that Elizabeth II's historical horizon extends no further than her father and grandfather, and some have jokingly referred to her as "George VII". There have on at least two occasions been rumours that Prince Charles, whose full name is Charles Philip Arthur George, may choose to reign as George VII rather than Charles III.
It might be noted that there has also been a previous "Prince George of Cambridge", namely the only son of Prince Adolphus, Duke of Cambridge (himself the seventh son of George III). This Prince George, who was a first cousin of Queen Victoria, was born in 1819 and rose to become Commander in Chief of the armed forces. He succeeded his father as the second Duke of Cambridge in 1850, but as he made a morganatic marriage the Cambridge title died with him in 1904.
George is also the name of the patron saint of England, on whose feast day, 23 April, the Order of the Garter is normally awarded.
The name Alexander has been borne by three Scottish kings, and was also the name of the youngest son of the future King Edward VII, who died at birth in 1871. Then there is of course Queen Alexandra, the consort of Edward VII, and Princess Alexandra, a first cousin of the current Queen of Britain and one of Prince William's godparents. Alexander is also the name of the only son of the current Duke of Gloucester.
The name Louis is most closely associated with Louis, the first Earl Mountbatten of Burma, a maternal uncle of Prince Philip and "honorary grandfather" to Prince Charles. Lord Mountbatten, who is known for his military career and for being the last Viceroy of India, was assassinated in 1979, and three years later the name Louis was given to Prince William, whose full name is William Arthur Philip Louis.
There is nothing particularly surprising about the choice of names, perhaps except that there are only three names. All the four children of Queen Elizabeth have four names, as have Prince William and his brother and the children of Prince Edward. However, Queen Elizabeth herself has only three names, as does the children of Princess Anne and of Prince Andrew.