A non-contact force is a force applied to an object by another object that is not in direct contact with it. This is observed with the magnetic force.
In all of the cases below there is no contact between the magnetic material and the object experiencing the magnetic force.
The force between two magnets
If one magnet if brought near to, but without touching, a second magnet it will exert a force on the magnet. As evidenced by the first magnet moving toward/away from the second magnet. A similar observation can be made when bringing a magnet near to a piece of a ferromagnetic material (iron, nickel or cobalt).
The magnetic force on a current carrying conductor
A current carrying conductor in a magnetic field will experience a magnetic force (perpendicular to the current and magnetic field directions), even though it is not in contact with the magnetic material itself.
The magnetic force on a charged particle
A charged particle moving through a magnetic field will experience a magnetic force which will cause it to move in a circular path.