If a person's documents don't match with the USCIS or SSA records, a 'tentative non-confirmation' comes up and it allows the employee several days within which he/she can rectify the information with the agency (USCIS or SSA). There are dedicated phone numbers to call and the onus is on the employee to contact the agency within the required amount of time. The employee is given the choice to not contest the 'non-confirmation' or to call the agency and get the situation resolved. If, after the days have passed, the situation is not resolved or the employee does not call the agency, then the employee can be terminated. Prior to that, however, the employee cannot be terminated and must be paid for the work done.
Additionally, E-Verify also has a photo match for immigration documentation, where a digital photo of the employee appears and it can be compared to the documents that the employee provides.
E-Verify is used in conjunction with the I-9 form, which is already a requirement for employers and requires to verify employment eligibility and identity, except that E-Verify can actually detect people who use fraudulent documentation.
E-Verify is not a republican thing and it's not a tool to deport anyone. Its purpose is to enforce laws that have been on the books for a long time - basically, it is an extra tool to make sure employees are lawfully authorized to work in the U.S. I have a green card and I've never been afraid of E-Verify, which I've used for employees ranging from U.S. citizens all the way to asylees and parolees who are authorized to work, and everything in between.
The error rate is less than 3%. And it gets better all the time. Moreover, no one who 'fails' E-Verify gets deported - they just don't get to have a job because the law clearly prohibits anyone who is not authorized for employment to legally work in the U.S. This law is not unique to the U.S. Most countries require that employees have the legal authorization to work.