If you are travelling outwith the UK with a child you must have the permission of any person with parental responsibility for that child - in most cases this means you need both parents' permission. To travel outwith the UK without the permission of a child's parents is technically child abduction. If you don't get the permission of those with parental responsibility for a child, you would need an order from the court.
Usually, depending on which foreign border you wish to cross, a letter from the left-behind parent or person with parental responsibility is sufficient to show that you have permission to take the child abroad. It is not unusual to be asked for written confirmation to prove that you hold such permission. Different foreign borders have different requirements on what they need to satisfy themselves that the child should appropriately be allowed entry into the destination country.
Remember that although a person is no longer considered a child at the age of 16 in the UK, it is not the same in every other country. Whilst a person of 17 would be considered a young adult in the UK, that is not the case everywhere else. You need to check the legal age at which a person is no longer considered a child in your destination county.
Letters of permission are not always requested upon entry to a foreign country, but the immigration officer at the point of entry is within his rights to ask for one. Ensuring that you pack all of the necessary paperwork along with your passport and foreign currency is the best way to ensure a stress-free holiday!