Can anyone explain how weights and masses may be compared using a balance

On one side of the balance you place the object of unknown weight. On the other side you place known weights (since gravity varies from place to place on Earth the know weights are usually labeled or stamped with their masses).
The object is to balance the balance such that the arrow points straight up at zero. To do this you start with the biggest known mass that will not balance the balance past zero. Then you add the next smaller mass that brings the arrow closer to zero without going past zero. Repeat with smaller and smaller masses until the arrow is as close to zero as you can get it. Finally you add up all the known masses and the total is the mass of the unknown object.
Some balances have a sliding weight (doctor's office) and are labeled in pounds (approximate) but they are still a form of balance scale.

It depends on the balance. Usually they are compared by observing the mass of each using a standard balance, a scientist saying "hmm", and carrying on with the experiment at hand.

One thing to keep in mind that you might be mixing up are the terms of 'weight' and 'mass'. The unit of weight is mass*acceleration. Weight is a force, such as a Newton or a pound. So the weight on earth of a 1 kg block is not the same of the weight on the moon of a 1 kg block. Mass is the amount of matter in an object, so mass will be the same regardless of the forces on it.