What is the normal force?

You've probably been shown bad explanations of what the normal force actually is or does. Under Newton's first law, an object can only remain stationary if no net force is applied to it. Say you have a brick on table. You assume the only external forces acting on the brick are gravity and the normal force. If only gravity acted, the net force on the brick would be downwards, and so it would phase through the table, which is of course unphysical. So, for the brick to remain stationary, there MUST be a normal force acting upwards to bring the net force to zero.

In this sense, the normal force can be thought of as a constraint force, in that it constrains the brick to move in a particular plane of motion, in that example the surface of the table. You don't require any information about the physical origin of the normal force to claim its existence, as the possibility of bricks phasing through tables is absurd. In this sense explaining it on the 'micro' level completely misses the point of the constraint force.

In Lagrangian mechanics, you essentially ignore constraint forces by reducing the number of physical dimensions of the problem. In general, a brick is able to undergo motion in 3 dimensions. On top of the table, the brick can only move in two dimensions. Each constraint force reduces the number of dimensions that need to be considered in the problem.

In simple mechanical problems you can assume these objects are indestructible. It doesn't make sense to talk about a 'threshold' with a constraint force. That being said, the physical origin of the force in the practical context can obviously always be overcome.

One other point, try not to confuse constraint forces with Newton's third law. The normal force represents a logical constraint on the system, Newton's third law represents symmetry. In the case of the table and brick example, the Newton's third law pair is not 'the gravitational force on the brick' and the 'force of the brick on the table'. This is an easy misconception in Newtonian mechanics where you visualise forces as being 'transmitted' through objects. Thinking about it that way will mean you cannot solve complex problems. The Newton's third law pair is 'the force of the brick on the table' and 'the force of the table on the brick'. The force of the table on the brick in this case is equal to the normal force but it is not the definition of the normal force.