The brackets work like you said, the first bit is taxed at the lowest amount, the next bit at the next amount, etc. So yes, only the part of your income that is in the highest bracket will be taxed at that rate.
You did leave something out though, at the bottom there is a portion of your income that is taxed at 0%. If you have ever heard people talk about personal exemptions, the standard deduction, and itemized deductions, all of these determine how much of your income is untaxed. For example a married couple with no kids gets 2 personal exemptions, 1 for each person in the household. Each of these is worth I think $3,900 for 2013. That part of your income isn't taxed. Then there's the standard deduction of $12,200 (or itemized if you have enough deductions to beat that) for a married couple, also not taxed.
Finally, after you figure out all of your tax, there's one more way to reduce what you owe - with tax credits. Some of the most common are the child tax credit ($1,000 off your tax bill for each qualifying child) and the earned income tax credit or EIC which is given to certain low income families.
Putting all of this together, for a family like mine in the 25% tax bracket, our total tax bill for 2013 was about 12% of our gross income. So, our income is definitely not all taxed at the top bracket, I think I would cry!