If we go back far enough, the titles of Prince and Princess were generally not used. Queen Elizabeth I was referred to as The Lady Elizabeth before acceding to the Throne, for instance. Regulation of a sort wasn't introduced into the British Royal Family until the accession of the Hanoverian Dynasty. With Queen Victoria there was still a certain flexibility and the present system was not confirmed until the great overhauls by HM King George V at the time of the First World War.
First of all, of course, we have the Sovereign. HM Queen Elizabeth II confirmed her desire that her heraldic crown should represent the Coronation Crown, named after St Edward the Confessor. Hence the heraldic crown she uses shows "s-bend" arches (as above) rather than what is generally termed a Tudor Crown which was introduced by Queen Victoria. This Tudor Crown (shown here) was used up until Her Majesty's father, King George VI. For the Scottish Coat of Arms, a closer representation of the Scottish Crown is used, including the pearl studs on the velvet lining. A distinction within British heraldry is that the Crown of the Sovereign only has 2 full arches which cross over in the middle. The Crowns of Continental Sovereigns generally have twice that amount.
Around the rim that supports these arches are Crosses and Fleurs-de-Lys. These have traditionally become a mark of Royalty, although HM King George IV expressed a wish for the Fleurs-de-Lys to be replaced by British symbols. He got his way with the State Diadem, the diamond headpiece that a queen generally uses on State occasions such as the present Queen to the State Opening of Parliament. The Crown remained unchanged with the somewhat French overtones of the Fleurs-de-Lys. The rim is further distinguished by jewels in traditional colours and the arches have pearls.
The next Coronet is that of the Heir Apparent. At the moment, this is HRH The Prince Charles who was created Prince of Wales in 1958 and invested when he was 21 in 1969. As such he is entitled to carry on his Coat of Arms a Coronet with one pearl-encrusted arch which is usually shown in full going from left to right. (Spain, for instance, shows a single arched Coronet side on, going from front to back.) Prince Charles was invested with a very modern interpretation of this Coronet in 1969 which was of pure gold after the first, electroformed, version disintegrated when the hallmarks were applied. As well as gold, it carries a ping-pong ball at the top which was heavily gold-plated to represent the orb. Strangely, though, the future King Edward VIII was invested in 1911 with a medieval-design circlet with no arch at all...
The next in rank would be other Princes and Princesses of the same generation (i.e. other children of a Sovereign than the Heir Apparent) who use Coronets with no arch at all but who retain the Crosses and Fleurs-de-Lys. (The heraldic term for these specific Crosses, by the way, is cross pattée, meaning that the arms of the Cross are narrow in the middle and splay out, more often in a curve, towards the ends.) Coronets with no arch have a gold tassel on top of the velvet, ermine-trimmed cap. Another distinction is that the rim is now and for further ranks (actually down to baron) "chased" instead of carrying actual jewels. This means that the rim has a pattern beaten into it to resemble jewels.
The next generation we come to is that of the grandchildren of a Sovereign. Here there are three possibilities, depending on lineage. The children of the Heir to the Throne replace the two Fleurs-de-Lys at the side with representations of Strawberry Leaves. This is the Coronet that TRH The Duke and Duchess of Cambridge and HRH Prince Harry use. The next rank of grandchild is the children of other Princes. Please note that, in general and unless specified otherwise, a person's rank is still taken from the father, hence, for instance, the children of HRH The Princess Royal do not rank as royal and do not have royal titles; their father was not raised to the peerage. otherwise, these grandchildren of a Sovereign carry a coronet of four Crosses Pattée and four Strawberry Leaves.
Please note that this rank no longer carries the style of Highness as opposed to Royal Highness.
With the Letters Patent which mean that all children of the eldest son of the Prince of Wales would be Prince/Princess and His/Her Royal Highness, not just the eldest son, and considering that HM The Queen might have inherited her mother's gene for longevity and the outside possibility that she might live to be 105, there is the equally outside possibility that she may still be alive for when Prince George turns 18 and therefore will have Arms assigned to him. He will be the first great-grandson of a living Sovereign to receive a coat of arms in modern times, although he will move up the ranks as time goes by. (The mess made with Prince Alistair of Connaught, Earl of MacDuff will be the subject of a later blog.) Prince George's Label may prove to be a problem, but only of design. The problem would be what Coronet to assign him!
A footnote to do with rank is the peculiarity to the British system of the princely title and the distinction made between the generations. The children and siblings of a Sovereign enjoy the style and title of His/Her Royal Highness The Prince/Princess. The Duke of Edinburgh was accorded this and as such is His Royal Highness The Prince Philip. Grandchildren of a Sovereign who enjoy this style and title (i.e. through the male line) drop the "The". Therefore, Princess Alexandra, the sister of The Duke of Kent, is styled Her Royal Highness Princess Alexandra. Prince William and Prince Harry will presumably move up the ranks as time progresses.