Newtons are a measure of force, so any system with a linear response could be used to measure. For example, a spring will stretch proportional to the applied force -- so you only need to make a scale on which to read how far the spring has stretched and calibrate it to Newtons.

Another method is to us a pulley to a known weight -- the rope will then exert a known force. If you hook the rope to the unknown force and then add weight until the rope does not accelerate, you can then read off the force by measuring the weight. For very small forces, one can use a magnet at a known distance from a piece of iron. If an electro-magnet (coil of wire)is used, the force is proportional to the current in the wire, which can be measured very accurately.