First, let's consider a few scenarios:
- A brick lands on your foot. Even though neither the brick nor the foot is moving, you feel pressure (and, well, pain, but let's ignore that for now) - the weight of the brick pressing against your foot.
- You accelerate in your car, and feel a force pushing you into your seat - the compression of the seat preventing you from passing through the seat.
- You jump off from a cliff to water. During the fall, you don't feel any force (though the whole experience can be disorienting and confusing, not to mention short, so you might not even notice). When you hit the water, you feel some force again as you slow down.
You can't feel the force of gravity, because it's uniform as far as your sensory equipment is concerned. Imagine your body as a spring - your sensors can tell you when the spring changes load (and gets shorter or longer). But under gravity, the same force "tugs" on your head as on your feet, so the "spring" keeps the same length, and has no net force.
However, things get more interesting where more forces get involved. You stand on the ground, which exerts a force on you that's the same magnitude as the gravitational force, but in opposite direction. This means that even though there are two forces acting on you, you experience no net force - the two forces cancel out.
Now, if the electromagnetic force that prevents you from falling through the ground was uniform, you wouldn't feel anything, and you'd just keep floating as if you were in orbit around the Earth. However, that's not the case here - the force gets stronger as you get closer. It's extremely strong in the area of contact between your feet and the ground (don't forget that it supports your entire weight against the pull of the entire planet), but it doesn't quite reach even into the skin on your feet. However, the skin of your feet is still tugged by gravity, so it wants to go down - only to press against the lower layers. It's this difference in forces that we can perceive - that's how you feel the ground beneath the feet.
A scale works in a similar way. In a typical mechanical scale, you have a static part (in contact with the ground), a spring of some kind, and a moving part you step on. The spring acts similar to the electromagnetic force (in fact, that's what ultimately drives it, but that's not important here) - the more compressed it is, the higher the force it exerts. So once you step on the scale, the spring compresses until it reaches an equillibrium - the force of gravity on your body exactly balances the force the spring (and the scale) exerts on your feet. At this point, there is no longer any (macroscopically important) acceleration - and yet, we can tell that the scale is slightly smaller than before you stepped on it. By understanding the way the spring is compressed under load, we can deduce how much load corresponds to a given compression of the spring.
But the main point in the explanation is the equillibrium. You only get a readout when the two forces balance each other out - that is, the sum of the forces acting on the scale adds up to zero, and there is no net acceleration.
And this is where F = ma comes together - you need to add up all the forces acting on a body to get the acceleration. When you start falling, there is little force opposing your fall, and the acceleration of your body is close to g. As you pick up speed, the air gets less capable of moving away from your path, and starts slowing you down - a force that acts against gravity, so your acceleration gets smaller, even though gravity is as strong as ever. Finally, if you fall far enough, the air resistance becomes so great, that it fully supports your weight against gravity - there is no more acceleration, the forces are balanced. But it's still an equillibrium - if you turned the force of gravity acting on you off for a few seconds, you would quickly slow down and eventually (in a few hours, probably) stop moving entirely.
But through all this, your weight is always the same - it's always exactly the force of gravity acting on your body, whether your body is supported (which makes you feel the weight) or not (the "weightless" feeling).