Contact forces occur when the two objects interacting with each other (exerting forces on each other) are physically in contact with each other (touching each other). Most forces are like this (for me to push on a box, I have to physically go up to the box, touch it, and push it).
Non-contact forces occur when the objects are not physically touching, but still exert forces on each other. The one everyone has the most experience with is the gravitational force. If I drop a ball, the gravitational pull of the earth pulls the ball down, even though the earth isn't physically touching the ball. There is a gravitational field created by the earth that pulls on the ball.
It is the same for electric forces between two charged objects (there is an electric field created that interacts with the other charged object) and with magnetic forces (there is a magnetic field created by a magnet that interacts with another magnet - or a moving electric charge). The two objects do not need to physically be touching, which is why it is called a non-contact force.