The force of weight ($W = mg$) acts on the apple. Who is applying this force?
Well, the Earth is.
And remember Newton's third law:
When one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body.
Then if a body A applies a force on a body B, body B will apply a force on body A such that they have the same magnitude and opposite directions.
What is important here? Well, Newton's third law talks about two bodies, A and B in my example.
If the Earth applies a force ($mg$) on the apple, the apple applies a force on the Earth with the same magnitude and opposite directions. See, weight is a force between the Earth and the apple, therefore its reaction must be acting on the Earth!
The Earth applies a force $mg$ on the apple.
Therefore, the apple applies a force $mg$ on the Earth. (Yes, the force of gravity is also subject to Newton's third law. The apple attracts the Earth gravitationally just like the Earth attracts the apple).
The normal force $N$ exists because objects don't tend to pass through eachother. This can be explained by electromagnetism, but for now, you just need to know that the contact between the hand and the apple causes electromagnetic repulsion which creates the normal force.
(In other words, the atoms in the hand repel the atoms in the apple, due to electromagnetic interaction).
Therefore, the hand applies a force (the normal force) on the apple, and the apple applies a force of reaction on your hand. That is why you 'feel' your hand being pushed when you hold something. You are applying a contact force on the object and the object applies a contact force on you!
A force and its reaction must be acting on different bodies. Remember A and B? Well, the apple and the Earth are different bodies, and so are the apple and the hand.
There are two kinds of forces in classical mechanics: field forces and contact forces. Gravity is a field force, and therefore its reaction must also be a field force (of course, it is also gravity). The normal force is a contact force, and therefore its reaction must also be a contact force. The force and the reaction have the same nature. Gravity and the normal force certainly don't have the same nature, and they can never be an action-reaction pair.
It turns out that in your example, $W$ and $N$ have the same magnitude. But this is just due to them being the only forces acting on the apple. And the apple is equilibrium! Therefore, Newton's second law, $F = ma$, guarantees that $N = mg$ so that the apple isn't accelerating and the net force is zero.
But this is just one case. It doesn't mean the normal and the weight will always have the same magnitude. They won't.
One example: the inclined plane.
You can see from the inclined plane's geometry that the resultant force on the object is $mgsinθ$, and it is constant. Therefore, by Newton's second law, the acceleration is also constant and equal to $gsinθ$.
And the normal force, always perpendicular to the contact surface, will have magnitude $mgcosθ$, and this is smaller than the weight's magnitude.
To finish things, a constant slope or constant inclination means that the inclined plane always (everywhere on its surface) has the same angle $θ$ of inclination relative to the ground.