Romney Fumbles, But Let's Examine His Claim Seriously
This Washington Post article explains the situation with 8 simple graphs, starting with Romney's claim and expanding from there to improve our understanding. 

Turns out that, Romney was focusing only on one particular type of federal tax.  In addition to federal income tax (where his numbers are accurate) there is the federal payroll tax (paid by many more) and there are state and local taxes (including sales taxes) that are more regressive (nearly everyone pays them) and hit lower-income groups harder. 

Adding up all taxes, the article shows that total tax rates are not that different across income levels.