What is the name of the paper that a doctor writes for someone letting his/her school, workplace, etc know that he/she has been sick for a period of time and needs to rest (off from work or school)?
It's called a medical certificate, or simply "doctor's certificate".
This certificate "attests to the result of a medical examination of a patient". Note that the result of the aforesaid examination can declare the examinee either fit or unfit.
The "sick note" is a "documentation that an employee is unfit for work", i.e., he/she is really sick.
In a private school in the UK, it might be called an exeat — Latin for “he may leave”.