Does friction depend on the area of the surface? Do the frictional forces vary from each other when different objects try to move on a surface? The materials, sizes, shapes are different from each other.
 A: It is the area of contact which matters, not the overall area of the surface something is moving on. If a ball of diameter 4 inches is rolling in a small room or a stadium, it makes no difference as long as the floor material and the speed of rolling is the same.
So if there are two bodies in motion (considering they are made of the same material, shape are moving at the same speed) then the larger body will suffer greater friction due to it's larger area of contact.
If the material and shape etc are different then everything comes into play. Friction does not depend on only one variable, but a whole bunch of them. In order to calculate the effect of one variable on the overall friction, you will have to keep all the other variables the same for two objects and then compare how the one you are changing, makes a difference.
A: The way in which we normally model friction is called Coulomb friction, this assumes that the frictional force is directly proportional to the applied load (the force pushing the two objects together- usually gravity).
When using Coulomb friction the area of contact is assumed to not effect the frictional force.
Coulomb friction is by far the most commonly used friction model and it is pretty accurate for most cases. However, it is inexact.
If you are interested in more detail like how friction changes when "The materials, sizes, shapes are different from each other" then I can give you some detail because I am writing a thesis on mathematical models for friction.
The materials involved can have various effects. For soft materials like rubber there is more adhesion (sticking) between the two surfaces, softer materials tend to stick to each other more so that adds to the friction. For harder materials like steel there is less adhesion but all the little bumps across the surface come in contact with each other causes a resistance to movement. In harder materials that resistance is higher when the hardness is greater. So the material is a complicated one, sometimes harder causes greater friction, sometimes softer causes greater friction
As for the sizes and shapes, I think what you mean is a bigger size has larger mass which increases the applied load at the surface contact. The different shapes would change the area of contact a bit, but if you have two rough spheres contacting each other then standard methods for evaluating the friction could be inaccurate.
