There were many competing measurement systems, at least one per country. Literally look up "pound" in Wikipedia and see the 4 systems used in England. After a while, there were conversions between the most-used systems and rules of thumb a out which system was best for which industry.

Generally, I believe they worked the same way the old standard Kilogram worked--there was one or a small number of reference units--this physical object is exactly one pound/mark/whatever. Standard weight sets were made using simple scales and comparing to the reference unit. And then the weights that merchants would use were made the same way, compared to standard sets (you don't want to handle the reference weight too much). Ideally, any two merchants could pull out a 1-ounce weight and they should balance out in a scale. And they largely did (though dishonest merchants might have a lighter set that just looked like a standard set to cheat people with. The Sheriff might compare weights, too, and then you might have to explain to the man with the keys to your jail cell why yours were so different....)

[ETA: for heavier objects you'd see scales with some mechanical advantage, but of a set amount. A "steel yard balance" is a good example. They were still balances--east to observe, hard to cheat, easy to replicate, as long as the weights were reliable]

Also remember that money was generally a set amount of precious metal. A silver penny contained X amount of silver and should weigh a set amount (ignoring clipping and devaluation). So, a payment could be literally weighed. But, it was still the same simple scales using known weights.