Proportional tax is a system of taxation where the amount to be paid as tax is a fraction of the income earned that does not change with the amount earned as income. For example, a if the rate of income tax is 10%, a person earning $50000, would pay $5000 as tax and a person earning $1000000 would pay $100000 as tax.
Progressive tax is a system of taxation where the rate at which income is taxed is not constant and usually increases as the gross income earned increases. The amount earned by a person is divided into different income brackets. The rate of taxation increases with increasing income brackets. For example, income earned till $50,000 may be taxed at 10%, income lying in the bracket $50,000 - $250,000 may be taxed at 20% and income above $250,000 may be taxed at 30%.
A progressive system of taxation is created on the philosophy that it is better to benefit those that earn less. As the marginal utility of income decreases with a higher amount earned, the increase in the rate of taxation for higher income brackets is justified. But taxation can be argued form the opposing philosophy that all income should be taxed at the same rate because earning a larger amount entails more time and effort. By this philosophy, it is therefore not right to increase the rate of taxation as income increases.