Weight

In common language, weight refers to the mass of a body; however, in physical terms it means the force on a body due to the earth's gravity, usually denoted by "Fg" or "Fw" or just "F" like any other force. It is equal to:

$$F_g = m\cdot g$$

Where g is a "constant" approximately equal to 9.80665 m/s2 at sea level at a latitude of 45.5°. However, g is inversely proportional to the square of the distance to the center of the Earth. Because of the oblateness of the planet (which results in a slightly greater distance between the earth's surface and its center, at latitudes closer to the equator) one is advised to go to the equatorial regions if one wishes to lose "weight."

At the equator, not only is one farthest from the center of the planet, but one also benefits from a centrifugal acceleration of 0.0336 m/s2. This is absent at the poles.

Weightlessness
You feel your weight when standing, sitting or lying, because the planet is pushing back at you with exactly the same force as that with which your weight is attempting to push the planet downwards (physicists call this the Normal force). Good thing, otherwise you'd drop through the planet.

If you're actually falling (as when skydiving, in the brief moment before air resistance begins slowing your fall) you feel weightless because there's no longer a planet pushing at your feet. Being in orbit is a special case of falling. You're falling toward Earth but with so much tangential velocity that you never get there. So you're weightless until Mission Control tells you to stop playing around and come home.

Note that just being at orbital altitude doesn't take your weight away. Imagine a really really long spiral staircase, with a landing at the top. As you climb, you progressively lose weight until, at the altitude of the Space Station, you weigh roughly five kilos less but you can still stand on the landing quite happily (as long as you don't need to breathe) watching the ISS whizz past.