Unless you are personally known to The Queen or another member of the royal family you should address your letter to the Private Secretary, an Equerry or a Lady in Waiting, asking for the subject of the letter be known to Her Majesty.
In such cases the letter can begin and end:
Dear Sir or Dear Madam (if name not known) Yours faithfully
Dear Sir Edward Yours sincerely
Envelope: The Private Secretary to The Queen (for example).
For those who do wish to write directly to The Queen, the letter should begin ‘Madam’ or ‘May it please Your Majesty’. It ends with ‘I have the honour to remain, Madam, Your Majesty’s most humble and obedient subject’ and then your name.
Within the body of the letter do not use ‘you’ (singular) or ‘your’ (singular). Instead use ‘Your Majesty’ and ‘Your Majesty’s’.
When being presented to The Queen men are expected to bow from the neck. Women can either bow or curtsey. If a curtsy is given it is just a short bob, keeping the back straight, hands by the side, dropping the knees slightly and bowing the head.
Those who do not wish to bow or curtsey do not have to. Technically, only those who have The Queen as their head of state need to bow or curtsey. Citizens of other countries, for example America, may bow or curtsey if they wish but do not have to.