Did you file a W-4 with your employer? You normally should have indicated you were single and had no other special adjustments. You should not have checked the box for "exempt" from taxes (this is rarely used and applies to people who are legally exempt, such as certain foreign workers.)
The first $15,000 of your income is not taxed, due to the standard exemption. The system for income tax looks as your wages as if you were paid the same thing for a whole year, and estimates the tax. For example, suppose you are paid weekly and your gross pay this week is $200. That's $10,400 over a year, which is less than $15,000, so no tax is withheld. Suppose next week you are paid $400. That's $20,800 over a year. After accounting for the $15,000 standard deduction, the taxable pay is $5800, the tax is 10% or $580, which divided by 52 = $11.15, and that would be your withholding.
Essentially, the system is self-adjusting. If you make less than the taxable threshold, you won't have withholding because you wouldn't owe tax at the end of the year. If you work more in some weeks, you may have tax withheld on those weeks. Then at the end of the year, all your income is added up, the tax determined (if any), and compared with your withholding. You would get a refund if your withholding is more than what you owe.
The only reason you might owe tax is if you had two jobs at the same time, because each W-4 will give you credit for the standard deduction but you only get that once. If you have two jobs at the same time, you need to file out the extra worksheets that are part of the W-4. But if you only have one job, and you did not check "exempt" on the W-4, then it should work out fine.
