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Tax brackets are the divisions at which tax rates change in a progressive tax system (or an explicitly regressive tax system, although this is much rarer). Essentially, they are the cutoff values for taxable income--income past a certain point will be taxed at a higher rate. The best explanation is an example.
Example
Imagine that there are three tax brackets--10%, 20%, and 30%. The cutoffs for these are $10,000; $20,000; $30,000. Under this system, someone earning $10,000 would be taxed at a rate of 10%, paying a total of $1000 (someone earning $5000 would pay $500, and so on). Meanwhile, someone earning $35,000 would face a more complicated equation. The rate on the first $10,000 would be 10%; the rate from $10-20 thousand would be 20%; and the rate above that would be 30%. Thus, they would pay $1000 for the first $10,000; $2000 for the second $10,000; and $4500 for the last $15,000; in total, they would pay $7500, or about 21.4%.
Some systems use an absolute tax bracket, where the entire amount is taxed at one rate. This can mean that someone earning just enough extra money to take them over the threshold to the next bracket, could pay even more than that extra in taxes.
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