Where does the employer get the information used to determine the amount of federal income tax to withhold from employees earnings?

From the IRS. They have instructions and pay charts telling employers what to withhold depending on how much the employee makes, how often they're paid, and what the employee entered on their Form W-4. There are also charts for certain types of work, such as agriculture. If you use this information to figure out if your employer is withholding the correct amount, don't forget to take into consideration any pretax benefits you have, such as a flexible spending account, which are subtracted before calculating tax.

States also provide similar instructions or pay charts for employers to determine how much state tax to withhold.

(See related link below for the IRS Pub 15 for employer instructions for withholding federal tax.)