Imagine that you're in an elevator. Sketch separate free-body diagrams for you, the elevator by itself, and the combined system of you plus the elevator for these three situations:
In this situation there are no new forces acting when there is an acceleration - one or more of the forces simply change size to produce the acceleration.
Your free-body diagram has two forces, the force of gravity and the upward normal force from the elevator.
The elevator's free-body diagram has three forces, the force of gravity, a downward normal force from you, and an upward force from the tension in the cable holding the elevator.
The combined system of you + elevator has two forces, a combined force of gravity and the tension in the cable.
Consider the normal force acting on you from the elevator:
The normal force is equal to your apparent weight. So, you actually feel a little heavier than usual when the elevator accelerates upward, and lighter than usual when the acceleration is down. In more extreme situations this is much more obvious. On a roller coaster, for instance, you feel very light at the top of loops, but heavier than usual at the bottom. The normal force applied by the seat on you is less than mg at the top and larger than mg at the bottom.