An inner function that keeps hold of variables from the function it was made in is called a closure: it "closes over" those variables. It's a way to bundle a little data with a function, without writing a class:

def make_multiplier(factor):
    def multiply(n):
        return n * factor
    return multiply

double = make_multiplier(2)
triple = make_multiplier(3)
print(double(10), triple(10))
print([double(n) for n in range(5)])

To change a remembered variable (rather than just read it), the inner function must declare it nonlocal, otherwise assigning to it would create a new local variable:

def make_counter():
    count = 0
    def increment():
        nonlocal count
        count += 1
        return count
    return increment

counter = make_counter()
print(counter(), counter(), counter())
other = make_counter()
print(other())

Each counter has its own private count, which no other code can touch.