Objects on the earth fall because there is a gravitation force between them and the earth - which results in an acceleration - as we saw above. So if you hold something in front of you and let it go - it will fall.
It falls because of an acceleration toward the centre of the earth which results from the gravitational force between the two.
These bodies move in a straight line from the point where they start to the centre of the earth. This means we can reuse everything we learnt in rectilinear motion. the only thing that needs thinking about is the directions we are talking about.
We need to choose either up or down as positive just like we had to choose a positive direction in standard rectilinear motion problems. this is the hardest part. If you can do rectilinear motion you can do falling body problems. Just remember the acceleration they feel is constant and because of gravity - but once you have chosen your directions you can forget that gravity has anything to do with the problem - all you have is a rectilinear motion problem with a constant acceleration!!
What is the weight of a free falling body ? Would you be surprised to know that it is zero ?
Weight is measured using a spring balance. (Mass is measured using a physical balance). When a body is falling, the spring balance to which it is attached (the spring balance needs to fall along with the body too !) would show zero weight. When the body stops falling, then it would exert a force on the spring (equivalent to the force exerted by earth on itself) and hence it would show a reading.