When you're a kid and you want to go to some event or a school excursion you have to bring a document signed by your parents saying that they've given their permission. What is this document usually called?
I know it as a "permission slip".
A permission slip in the United States is a form that a school or other organization sends home with a student to a parent in which the parent provides authorization for minor children to travel under the auspices of the school or organization for some type of event, such as a field trip.
Depending on the situation, it could be a waiver if it involves agreeing to some element of risk.
A waiver is the voluntary relinquishment or surrender of some known right or privilege.
It the UK it depends on context and content:
Permission and consent are interchangeable to allow someone to do something. Generally permission would be to allow a minor to take part in something, consent has a slightly more formal contractual association.
Authorisation to allow someone to do some act, or incur something on your behalf.
Waiver or disclaimer if you are giving up a right (usual to claim or sue).