What is coefficient of friction ?

Simply put, it is a measure of the force required to move one object over another. A car is moving along a road smoothly. It hits a wet patch and skids. This is because the friction between the tyres and the road suddenly decreased. Every one laughs at a person, who takes a toss slipping on a banana peel. The toss is a result of reduced friction between the leg and the road. In engineering there is a need for quantifying this value as it figures in all motion design.
Consider a weight on the ground. It is acting vertically down. Now if a horizontal force is applied to the weight to move it, there is a resistance till the force exceeds a certain value. The ratio of the force needed to move divided by the weight is called the coefficient of friction. If a smaller object of the same material is used, the force will also be less.
However, the Ratio (and hence the coefficient of friction) will still be the same.

Since the two forces are vectors, the resultant of the two vectors is another vector at an angle to the line of action of the weight. The ratio happens to be the tangent of this angle.