An everyday answer is: It is a force that acts against motion.

When two objects touch, and you try to slide them along each other, you must apply enough force to get the motion to happen.
If you push on a desk, it won't move until you reach some threshold force level.

This type of friction is called static friction, and exists when the objects don't move.

While you push and the desk sits still, the force of friction EXACTLY MATCHES your applied force.

There will be no motion as long as this equation is true:  Fapplied < Fstatic friction

What does friction depend on?

It depends on (1) the surfaces in question and (2) on how much external force there is holding them in place...

(1)Different surfaces display different levels of friction... It is harder to slide a box on cement than it is on ice.  This rating is called the friction coefficient, and is represented by "m".  What unit is the coefficient measured in?  None!  If it takes 3N of pushing force to barely budge a box that resisted with 2.9N,  any ratio of these two numbers cancels out the "N".  A high coefficient means it takes more force, a lower coefficient means it takes less.

(2)For a box sitting on the floor, gravity provides the force mg acting downward, which is opposed by the normal force acting upward.  For static friction, the frictional force is given by:  Fstatic friction < mN
 

How much force will it take to just budge a 40 Kg box sitting on a cement floor if m=.4?
USE: Fstatic friction < mN
Remember N = mg
mN = mmg = .4*40Kg*9.8m/s/s = anything above 156.8N will move the box