Gravity is pulling on you downwards with a force which we call your weight. The reason why you aren't accelerating downwards is that there is an equal and opposite force of the floor (let's assume you are standing up) pushing you upwards that nets out against the force of gravity. This is the "equal and opposite" force described by Newton's Third Law of Motion.?

This force, that avoids that our body be accelerated downwards in spite of the attractive force from the earth, is called the normal force, as it is always normal to the surface on which the object is placed.

It is not the "equal and opposite force" described by Newton's Third Law of motion, because, this law says that this force acts on both bodies simultaneously, ( as a pair) so, one force is the one exerted by Earth on the body (which we call weight) and the other is exerted by us on Earth (acting at the center of the Earth, upward).

Also, the case when the normal force is equal and opposite to gravity force, is a special case, when the surface is level, and the object is at rest in the vertical direction.

Normal force can adopt any value so the combination of external forces be compliant with Newton's 2nd Law.