Unfortunately, the tax system in the U.S. is probably more complicated than it looks to you right now.
First, you need to understand that there will be taxes withheld from your paycheck, but the amount that they withhold is simply a guess. You might pay too much or too little tax during the year. After the year is over, you'll send in a tax return form that calculates the correct tax amount. If you have paid too little over the year, you'll have to send in the rest, but if you've paid too much, you'll get a refund. There are complicated formulas on how much tax the employer withholds from your paycheck, but in general, if you don't have extra income elsewhere that you need to pay tax on, you'll probably be close to breaking even at tax time.
When you get your paycheck, the first thing that will be taken off is FICA, also called Social Security, Medicare, or the Payroll tax. This is a fixed 7.65% that is taken off the gross salary. It is not refundable and is not affected by any allowances or deductions, and does not come in to play at all on your tax return form.
There are optional employee benefits that you might need to pay a portion of if you are going to take advantage of them, such as health insurance or retirement savings. Some of these deductions are paid with before-tax money, and some are paid with after tax money.
The employer will calculate how much money they are supposed to withhold for federal and state taxes (yes, California has an income tax), and the rest is yours.
At tax time, the employer will give you a form W-2, which shows you the amount of your gross income after all the before-tax deductions are taken out (which is what you use to calculate your tax). The form also shows you how much tax you have paid during the year.
Form 1040 is the tax return that you use to calculate your correct tax for the year. You start with the gross income amount from the W-2, and the first thing you do is add in any income that you didn't get a W-2 for (such as interest or investment income) and subtract any deductions that you might have that are not taxable, but were not paid through your paycheck (such as moving expenses, student loan interest, tuition, etc.) The result is called your adjusted gross income.
Next, you take off the deductions not covered in the above section (property tax, home mortgage interest, charitable giving, etc.). You can either take the standard deduction ($6,300 if you are single), or if you have more deductions in this category than that, you can itemize your deductions and declare the correct amount.
After that, you subtract more for exemptions. You can claim yourself as an exemption unless you are considered a dependent of someone else and they are claiming you as a dependent. If you claim yourself, you take off another $4,000 from your income.
What you are left with is your taxable income for the year. This is the amount you would use to calculate your tax based on the bracket table you found.
California has an income tax, and just like the federal tax, some state taxes will be deducted from your paycheck, and you'll need to fill out a state tax return form after the year is over to calculate the correct state tax and either request a refund or pay the remainder of the tax. I don't have any experience with the California income tax, but there are details on the rates on this page from the State of California.