It could happen. You walk into a pub and there's Prince Harry. You attend a garden party and there's the Queen. In an attempt to help you keep your composure, following is royal meet & greet protocol. Try to follow it so as not to make all Americans look hopelessly backward.
How to bow or curtsy:
Men: Bow from the head only. This is called a neck bow.
Women: Do a small curtsy. This video shows proper execution.
Or: Simply shake hands in the usual way.
How to address members of the royal family:
The Queen: The correct formal address is to say 'Your Majesty' the first time, and then simply 'Ma'am' after that. Please note that ‘Ma’am’ does not rhyme with ‘jam.’ Say the American word 'mom' and you'll be close enough.
Male members of the Royal Family: use the title 'Your Royal Highness' first, then 'Sir'.
Other female members of the Royal Family: use 'Your Royal Highness' followed by 'Ma'am.'