While there are intricate rules for which royals must bow or curtsy to other royals, the rules for the public are surprisingly simple. It is not required for members of the public to bow or curtsy to royalty, although many still do it out of respect.
"There are no obligatory codes of behaviour when meeting The Queen or a member of the Royal Family, but many people wish to observe the traditional forms. For men this is a neck bow (from the head only) whilst women do a small curtsy. Other people prefer simply to shake hands in the usual way," the official royal website explains.