A tax exemption is an amount of money that can be deducted from one's gross income when figuring out the amount of money on which tax will have to be paid.
Your gross income is the total amount of money that you make. But you do not typically have to pay taxes on all of that money. Instead, there are certain exemptions that you can take to reduce the amount of income that you officially have. For example, there are tax exemptions for each person who is a member of your household. You get to take the amount of the exemption, multiply that by the number of people in your household, and then deduct that product from the amount of income that you have to claim.
By taking exemptions like this (or like those for charitable giving), you can decrease your taxable income and thereby have to pay taxes on a lower amount of income.
I completely agree with the above answer. Pohnpei397 started talking about the definition of what an exemption is and just began to mention the different types of exemption that there are. The person mentioned that one can get an exemption for members in their family. This is true. One can get exemptions for the children that they support but they must be able to provide the child's (ren's) social security numbers. One can also get exemptions for the amount of money paid for child care if one can provide receipts documenting the amount that was paid. One can also get an exemption for the amount of money paid for interest on a home or on credit card debt. Those are just a few of the items that one can get exemptions for.