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.
Imagine that there are three tax brackets: 10%, 20%, and 30%. The 10% rate applies to income from $1 to $10,000; the 20% rate applies to income from $10,001 to $20,000; and the 30% rate applies to all income above $20,000.
Under this system, someone earning $10,000 would be taxed at a rate of 10%, paying a total of $1,000. Someone earning $5,000 would pay $500, and so on.
Meanwhile, someone earning $25,000 would face a more complicated calculation. The rate on the first $10,000 would be 10%; the rate from $10,001 to $20,000 would be 20%; and the rate above that would be 30%. Thus, they would pay $1,000 for the first $10,000 of income (10%); $2,000 for the second $10,000 of income (20%); and $1,500 for the last $5,000 of income (30%); in total, they would pay $4,500, or an 18% average tax rate.