Just a minor, but important, detail to add to TopSquark's post...

The frictional force always has the following properties:

1. It always resists motion, it never aids it; and

2. It will always equal the pushing force, up to some limiting value.

The formulae provided by Topsquark are the limiting values. In general, \(\displaystyle f_x \le \mu N\).

To see how this might be important, let's consider a simple example. Consider a box with a weight of 1000N on a horizontal surface with a coefficient of static friction equal to 0.5. Due to Newton's first law, the normal force is equal to the weight.

The limiting frictional force is then:

\(\displaystyle f_x = \mu_{max} N = 0.5 \times 1000 = 500\)N

However, if someone pushes horizontally on the block with a force less than this, say, 200 N, the frictional force will equal it (200N) and will resist the pushing force. The forces are in equilibrium and the box will not accelerate.

Here's a graph showing how, in general, the frictional force varies with applied force.

As the applied force is increased, the frictional force increases to match it. Eventually, however, the applied force starts to exceed the limiting value for static friction (point B). For applied forces above this limit, the object will accelerate. The frictional force then dips slightly to the dynamic limit (C) because of the lower coefficient of resistance.