|HRH Prince William Duke of Cambridge|
Queen Elizabeth II is head of state of the United Kingdom of Great Britain and Northern Ireland, Canada, Australia, New Zealand, Jamaica, Barbados, the Bahamas, Grenada, Papua New Guinea, the Solomon Islands, Tuvalu, Saint Lucia, Saint Vincent and the Grenadines, Belize, Antigua and Barbuda, and Saint Kitts and Nevis (collectively known as the Commonwealth realms). Added to these independent states are the Crown Dependencies which include the Isle of Man (where the Queen traditionally holds the title of Lord of Mann) and the Bailiwicks of Jersey and Guernsey and the Royal Fiefs of Alderney and Sark which collectively are commonly referred to as the Channel Islands (where the Queen traditionally holds the title of Duke of Normandy).
I have listed all these states and territories to illustrate how complicated it will be to change the current laws governing the succession. Under the current law, the oldest son inherits ahead of any younger brothers or any older or younger sisters. Only if there is no son does the oldest daughter inherit. Prince Philip as the husband of Queen Elizabeth is the Prince Consort and not in line for the throne.
Succession is usually the story of the heir and the spare. While the role of the heir is clearly cut out, the spare is more or less just there in case something untoward happens to the heir. While the spare enjoys all the privileges of a major Royal, he has to be able to accept that his descendants are relegated to minor Royals with no future in the Firm. Prince Andrew as Charles‘ spare has just been told in no uncertain terms that his daughters Princess Beatrice and Princess Eugenie are not worth 24 hour protection and that they should start looking for a career outside the Royal Family.
The current heir to the throne is Prince Charles, Prince of Wales, Duke of Cornwall, Duke of Rothesay, Earl of Carrick, Baron of Renfrew, Lord of the Isles, and Great Steward of Scotland. In the order of precedence he comes third after his parents in the United Kingdom; in other countries he comes fourth or fifth after the Queen, the vice-regal representative(s), and his father. He is married to Camilla Parker-Bowles who chose to be known as Duchess of Cornwall rather than as Princess of Wales like her predecessor (but she effectively still holds rank and title).
Second in line is Charles’ older son Prince William, Duke of Cambridge, Earl of Strathearn, Baron Carrickfergus. In the order of precedence, he comes after his father but before his stepmother. He is married to Catherine (Kate) Middleton. The complicated order of precedence for the Duchess and Cornwall and the Duchess of Cambridge are explained in The Duchess of Cambridge: Curtsy in a Muddle.
Third in line to the throne is William’s younger brother and spare Prince Harry. After him come Prince Andrew, Duke of York, Earl of Inverness, Baron Killyleagh, and his daughters Beatrice and Eugenie. After them come Prince Edward, Earl of Wessex, Viscount Severn, and his children James Viscount Severn and Lady Louise Windsor. Finally from the Queen’s children comes Princess Anne, The Princess Royal, and her children Peter and Zara Phillips (Anne, ever the realist, refused any hereditary titles to allow her children to fade out of the Royal Family).
The compilation above shows several things and should answer several questions. A single person may be graced by several titles. While it is usual to use the title that is highest in precedence (like Prince of Wales) a person may chose to use a lesser title (like Duchess of Cornwall). It is also possible to use a local title (like Baron Carrickfergus) when travelling in Northern Ireland to emphasise the national aspect.
As to the requirement of a bow or a curtsy, they are reserved for formal occasions. While subjects of the Commonwealth realms and the Crown Dependencies are expected to use the formal address, backstabbers like the Prime Minister of Australia might prefer to show their lack of manners, breeding and education by not doing it. Citizens of other countries are not expected to either bow or curtsy but may do so out of respect for the person rather than the institution (meaning that Americans will not have to bow or curtsy to William and Kate).
With the birth of Prince George of Cambridge, we have a new third in line to the throne. Prince Harry and everyone after him are moving one step down on the ladder. The governments of Her Majesty's realms have also made a constitutional change to the succession for all descendants of William and Kate. In their bloodline, it doesn't matter if the child is male or female to inherit, the first born will come first in the succession. As this doesn't apply to all other bloodlines, keeping track of the rights of succession became just that much more interesting.
How Many Monarchies Exist in Europe?
William and Kate: Title History
Prince George, Duke of Cambridge