The terms progressive tax and regressive tax describe taxes that affect different levels of income differently. They don't refer to political ideology.
A progressive tax is one that taxes high wage earners more than low wage earners.
A regressive tax is one that taxes low wage earners more than high wage earners.
A flat tax is one that taxes everyone equally.
In the following examples, I describe people as "poor" or "rich" to increase the contrast between the types of taxes.
Examples of progressive taxes:
Income taxes. The more you earn, the more your income is taxed. Right now, the tax rate ranges between 10% and 38%, with the highest paid people paying the 38% rate. Of course, they pay 38% only on a fraction of their earnings, which is normal for a progressive tax. See tax brackets for clarification.
Capital gains taxes. Though this is generally a flat tax, it's wealthy people who tend to have the most capital gains, because they own more assets.
Examples of regressive taxes:
Sales taxes: poor people buy things, and save almost nothing, thus most of their income is taxed. Rich people buy things, maybe more things, but save a lot of money (or invest it), and that saved money is not subject to sales tax. Poor people end up paying more of their income in taxes.
Parcel taxes: poor people who live on small parcels pay the same as rich people who live on large parcels.
The "flat tax"
It's actually difficult to create a flat tax. The sales tax is nominally a flat tax, but in reality, it operates in a regressive way.
The typical flat tax proposed by politicians usually includes a large exemption from taxes for income below a certain amount, like $20,000. So, it really starts out as a progressive tax.
Then, there's a range in there where the tax really is flat. As income rise, however, there are ways to avoid taking your pay as "income", usually through benefits and stock options. Since that tax is now dodged, the flat tax is no longer really flat.
It's progressive for the poor, and regressive for the rich.