GCD - Base Case

When does the recursion stop? When $b = 0$. At that point, $\gcd(a, 0) = a$. Any number divides $0$ (since $0 = 0 \times n$ for any $n$), so the GCD of $a$ and $0$ is just $a$ itself.

This is your base case. Every recursive call reduces $b$ (since $a \bmod b < b$), so you will always reach $b = 0$ eventually. This is where recursion stops. Without it, you get infinite recursion and a stack overflow. Every recursive solution needs at least one base case.