PPM stands for parts per million. Its fairly synonymous with mg/L (milligrams per liter in liquid solutions) or mg/Kg (milligrams/Kilogram in solid mixtures).
Measuring "it" depends on the chemical in question. Since we were talking about N (nitrogen), you need a nitrogen specific test to measure it. The are several methods to measure nitrogen from either the ammonia, nitrate, or nitrate concentrations, and I believe urea N is measured as TKN (total Kledjal Nitrogen). You can get cheap and dirty ammonia and nitrate test kits from an aquarium store. But typically we just guesstimate the N in a fertilizer solution by straight math based on the percentages of the different ingredients in the fertilizer.
http://pubs.ext.vt.edu/430/430-100/430-100.html
Try the above link for a tutorial Jack
Rick: ammonia is also measured by the Kjeldahl method, we measure nitrate (by difference) before and after its conversion in ammonia after its reduction. Urea alone is measured after its hydrolyse (in ammonia) by .... urease.