So, part of doing your taxes in the US is figuring out the actual amount of income that will be used to calculate the tax. This isn't just the sum of all the money you made; you are permitted to also figure up how much money you spent on certain types of expenses, and subtract that amount from your income. The permitted types are things like medical bills, payments of other types of tax, etc.
So suppose someone had $60,000 income, and $5,000 in expenses which fall into the permitted categories; they could calculate their tax as if their income was $55,000.
But to do this, you must use what is called "itemized deduction"; you must be able, if asked, to provide documentation of the amount of each expense and proof that it was of a permitted type.
Or you can use the "standard deduction". This is a simple flat amount that anyone can apply with no questions asked. For example, if the standard deduction was $6,000, the person above with $60,000 income could simply apply the standard deduction and pay tax on income of $54,000, and not have to document any specific expenses.
The exact amount of the deduction depends on whether you are single or married, have children you pay to care for, etc.
Some people spend a lot of time figuring out which option will result in the lowest amount of tax; most people just use the standard deduction unless they know that they have enough large expenses to be worth the trouble of itemizing.
For some people with low income, this means they pay no tax at all, or even receive a refund -- in the US, employers have you fill out a form stating how you plan to file your taxes (i.e., if you will be applying deductions for being married/having children/etc.) and automatically withhold from your pay and send to the IRS an amount calculated to match what you will probably owe. If the amount withheld is greater than the amount of tax you actually needed to pay, you receive money back. Some people with higher income also deliberately have their withholding set high; this is based on the idea that it is better to withhold too much and receive a refund than it is to withhold too little and perhaps have an unexpectedly large tax bill that must suddenly be paid.