Tax Rate, Loyalty, and Public Grievance are three numbers which are closely intertwined. The three will always attempt to total 100, and there are different ways to affect each of them.Your Tax Rate determines how much gold you make per hour, based on your entire population (including unemployed.) Tax Rate is the easiest number to manipulate, the panel for which is found in the Town Hall. Tax Rates are affected only by Tax Policy and Adv. Tax Policy.
Loyalty affects how many people are in your city. If you have 50 Loyalty, you will have 50% of your maximum population. Loyalty adjusts itself based on your Tax Rate and Public Grievance. In addition to being affected by Tax Rate and Public Grievance, it is affected by Disaster Relief and Praying.
If captured, an opponent's Hero can be persuaded to join your forces if plied with Medals and gold. This can also occur vice versa, in which you can lose an attacking hero whose loyalty has dropped to 0.
Public Grievance is a number whose purpose is to stymie the growth of Loyalty. It does not do anything but act as a "filler" of numbers where you could instead have higher Loyalty or Tax Rate. It is affected by losing attacks to your city, as well as Disaster Relief and Praying.