A tax is defined as regressive if the proportion of income paid as tax falls as income level increases.
In other words, a regressive tax takes a larger percentage of income from low-income earners than from high-income earners. This means that the burden of the tax falls more heavily on those with lower incomes, as a larger portion of their income goes towards paying taxes.
For example, a sales tax that applies the same tax rate to all goods and services regardless of their price, will take a larger proportion of a low-income earner's income than a high-income earner's income, since the low-income earner must spend a larger portion of their income on these goods and services.
A progressive tax, on the other hand, is one where the proportion of income paid as tax increases as the income level increases. This means that higher-income earners pay a larger percentage of their income in taxes than lower-income earners, thus reducing the tax burden on low-income earners.
A proportional tax is one where all income groups pay the same percentage of their income as tax. A fixed nominal amount of income tax for all income groups is a flat tax.