Why is the force of gravity not equal to the normal force on an inclined plane? When an object is at rest the force of gravity and normal force cancel out. However, when place on an inclined plane the downward force of gravity is not equal to the normal force that is perpendicular to the ramp. Why is that?
My second question is why is normal force perpendicular to the ramp?
 A: Because you are forgetting friction.
If there is no friction then the force of gravity on the box will be greater than just the normal force from the plane - hence the forces won't cancel out and the box will slide down along the inclined plane. If you add friction to the surface between the box and the plane then if the friction is great enough it will make a force upward and along the plane that adds to the normal force from the surface and the box will be at rest.
The normal force will always act perpendicular to the surface. You can imagine that it is a force preventing the box from going straight through the plane. If it wasn't acting perpendicular then the box would not slide downward along the plane but rather move in some peculiar (and magical) way.
A: Even without friction, the normal force is defined as being perpendicular to the surface. If the surface is tilted, the normal force cannot be equal and opposite to gravity.
A: The net force on a body is $\mathbf F = m\mathbf a$. If friction is small for example, an object can slide along the inclined plane with an acceleration. The net force has the same direction.
But another concept of force relates it to elastic deformation instead of acceleration: $\mathbf F = k\mathbf x$. A perfect cubic object, lying in the ground by its own weight, has the vertical dimension smaller than the other two due to that elastic deformation.
If it is lying on a wedge, there is a reduction in the dimension in the normal direction ($\mathbf N = k\mathbf x$), and an angular deformation. The later is a measure of the couple caused because the friction force is not applied in the center of mass.
The forces can be measured by strain gages attached to the cube. Small deformations produce electric signals proportional to them. It is the principle behind most load cells.
A: One more thing to note, is that normal force is force of constraint.
We know, that the object cannot move under the surface and there must be some force that prevents this from happening. Such force must be perpendicular to the surface, because it only prohibits things to go under it and does not influence anything else. This is how normal force is defined.
But of course, the true contact force from the ramp might be more complicated than that. We call this additional complication friction.
