Those on low incomes would almost certainly pay more because around the world flat tax systems are associated with high National Insurance contributions - that hit the lowest paid hardest. So flat taxes are really about cutting taxes for the best off, cutting services (like the NHS) massively and requiring payment for their use instead, and increasing tax, overall, for the least well off. That's the reality. (...) So flat tax would simplify almost nothing, but leave you paying to see the doctor or educate you children. That's what the flat taxers fail to mention. 
A flat tax, also called a fixed rate tax, as well as other things similar to that, is a fixed tax rate which is applied to the population of a country/area as a whole, regardless of a resident's annual pay or living situation. For example, a mother working two jobs to keep her kid in school while living in a tiny council house which only has one toilet (possibly right next to the kitchen and therefore pretty gross) would be paying the same proportion in tax as a multi million-billionaire living in a massive country house with a racing car and a sexy wife where he has like 3 toilets and only one of them is in one of his 2 kitchens.
Due to the scathing example provided above, it's fairly obvious that a flat tax is an example of right wing financial philosophy.