The normal force isn't fixed in magnitude. It adjusts its value according to other forces in the system. To calculate the value of the normal force, calculate the sum of the forces perpendicular to the surface and set that equal to $$0$$.
For example, in this case, this becomes $$F + N - m g = 0$$, where $$N$$ is the normal force exerted by the surface and $$F$$ is another external force.
$$F$$ and $$N$$ act upwards, so, they have positive signs in the equation. The weight acts downwards, so, it has a negative sign in the equation.