The purpose of unit tests is to verify that you understand what an individual unit of your code does at the moment that you write it.
A "unit" is a function or a method; some part of your program that has a discrete amount of behavior.
That is the only purpose of a "pure" unit test, and it is possibly the most important aspect of testing, although not the only important aspect of testing.
As a labor-saving measure, you can have unit tests serve other purposes as well.