What is gravity?
Every object and every celestial body, such as planets, moons or asteroids, exude their own force of attraction, called gravity. The magnitude of this force is dependent on the mass of the object. The heavier the object the stronger is the pull of gravity.
What is weightlessness?
A state in which we cannot feel the effect of gravity is called weightlessness or to be more precise "microgravity". This is the case when an object falls to the ground unimpededly. It is in free fall, because there is no other direct force affecting the falling object.
If we attached a scale to our feet before we jump from a diving platform, it would show virtually no weight at all, except for the force of the air drag that is exerting to the scales. We can observe the same effect after an object has been accelerated and is now flying upwards, e.g. when we have been thrown into the air by a trampoline. In this case we are in free fall on the way up as well as on the way down. If we were now able to eliminate the air drag and perform the free fall in a vacuum, we would achieve a far better quality of microgravity.
In the Bremen Drop Tower we are using exactly this effect: Due to the vacuum inside the drop tube we can achieve a quality of microgravity of one millionth of the Earth's gravity in free fall.