Gravity

Why do planets, asteroids and comets stay on the same path as they travel round the Sun? Why does the Moon keep travelling round the Earth instead of flying into space? The answer is gravitation or the force of gravity.
All objects exert a force on each other. The bigger the mass of the object then the bigger the force. This is an attractive force called gravity. This force causes objects to be attracted to each other. If an object has a mass the size of a planet - then it will attract all sorts of things to it (people, cars, elephants, even moons). This is why we can jump up in the air away from planet Earth - but we can't jump very far, and we are attracted straight back down to it instead of floating off into space.

Orbits

The reason why objects move in an elliptical (curved) path around a planet or star was first explained by the British scientist Isaac Newton in 1728. He described it like this: imagine an extremely powerful cannon. If it fires a cannonball then the ball will travel through the air but it will eventually fall to the Earth due to the force of gravity (as shown by the blue line in the diagram).
Now, imagine you increase the power of the cannon. The cannonball will travel away from the cannon before starting to fall towards the Earth. If the cannonball has just the right amount of energy it will miss the Earth (because the Earth is curved) and carry on falling. But, because the Earth is curved, the cannonball will keep missing the Earth as it falls and will come round to the starting point in a circle - or orbit (the red line in the diagram). This is how the Moon (and man-made satellites) orbits the Earth. It is constantly falling towards the Earth but just missing it - because it is travelling at the right speed.
If the cannon is given even more power then the cannonball will have enough energy to escape the gravitational pull of the Earth and will fire into space (shown by the green line in the diagram). When we send a rocket into space they have to be travelling fast enough to achieve an escape velocity which means they have enough energy to overcome the gravitational pull.

Weight

Weight is a force (measured in Newtons). It is a measure of how much something is attracted by the force of gravity. For example, my body has a certain mass (measured in kilograms). If I stand on some scales on Earth then the Earth's gravity will pull my mass towards it creating a force. This force is what we often call 'weight'.
If I went to the Moon and stood on some scales then my 'weight' would be different - even though my mass hadn't changed. This is because the Moon has less mass than the Earth, so it would produce less of a pull on me than the Earth would. In other words, the force of gravity on the Moon is smaller than on the Earth.
If I went to Jupiter, though, the pull would be so great that I wouldn't be able to stand up on the scales anyway!