remember that normal force exists because it is the force that keeps an object at rest(assuming it actually is not moving!), despite gravity pulling down at mG.
One way to calculate the normal force is to solve for it. If you know that F=ma and a = 0 , then do a free body diagram and summate the forces to equate to 0.
If it IS moving as in circular motion if you know the acceleration, you know the net force. (recall that in uniform circular motion the acceleration is the centripetal force or v^2/r.
Therefore you can calculate the normal force of a car traveling on a hill or in a ditch, by summating the Normal force, gravity (does not change), and equating it to mv^2/r (v and r should be given, as should m)
If the problem describes motion on the end of a string, you also include tension, but I can't think of an example where you have an object resting against something AND have it attached to a string. So basically it reduces to
T-mg=mv^2/r