The “Flat Tax” is a variant of an Income Tax that taxes every dollar of income at the same rate. For example, after determining your taxable income, that amount would be taxed at, say, 25%. Rich or poor, every dollar of taxable income is taxed the same. In some cases (I have never seen an exception), the Flat Tax begins with an exemption of the first $20,000 or $30,000, to soften the blow on low wage earners.
The “Fair Tax” is another variant of an Income Tax that is really a fixed-rate Sales Tax or Consumption Tax. For example, you do some shopping at your local Big Box store and end up with a big wagon of stuff; in addition to local and state sales taxes on individual items, there would be a fixed rate federal Sales Tax on the total checkout that would be the same in every locality in the nation, and this would replace the old Income Tax. And like the Flat Tax, the Fair Tax begins with what they call a “prebate,” a monthly check, to soften the blow on low-wage earners (and everyone else).