This is, of course, a matter of opinion. I would argue that the American tax system is basically fair but that it needs some changes to be fairer.
The main reason that I think the tax system is fair is because it is generally progressive. Our income tax rates on the national level are progressive. I believe that it is fair that people who have a greater ability to pay should pay a higher percentage of their income. Such people benefit greatly from what our society has to offer and it is right that they should pay to help support the system. While they do have to pay more than people lower down the economic ladder, this is just because the poorer people simply would not be able to afford to pay much more.
The system could, however, be fairer. Poorer people pay a larger proportion of their income to other taxes such as gas taxes, sales taxes, and Social Security taxes. It would make sense to reduce the burden of such taxes, particularly Social Security taxes, on people with lower incomes.
Of course, it is easy to argue the other way. There are many who believe that a flat tax is the only fair tax since it taxes all people equally. I, however, believe that some amount of progressivism in our tax structure is desirable.