Directly from the USCIS website:
All U.S. employers are responsible for completion and retention of Form I-9 for each individual they hire for employment in the United States. This includes citizens and noncitizens. On the form, the employer must verify the employment eligibility and identity documents presented by the employee and record the document information on the Form I-9. The list of acceptable documents has been amended in the 2007 version of the Form I-9 and can be found on page 4 of the forms.
The link takes you to the USCIS website's I-9 Form.
The previous answers are absolutely correct, but I just want to clarify something given the wording of the question: what the employer must verify on the I-9 form is the eligibility to work of each new employee, NOT the citizenship. The employer can not insist on a specific immigration status (including U.S. citizenship or permanent residence) without violating various discrimination provisions, and should simply supply the lists of acceptable documents on the back of the I-9 form to the employee and let the employee choose what to present. As long as the employee presents documents showing either identity as well as eligibility to accept employment (List A documents as listed on the form) OR an identity document (List B) plus a document showing employment eligibility (List C), the employer must accept this regardless of the immigration status indicated. The only time an employer could ever insist on citizenship status would be where it was a legitimate requirement of the job, such as certain military contract work might require.
I-9 record keeping and reporting has been lax in the past, but because of stepped up ICE raids on employers who routinely hire undocumented aliens, coupled with the hostile political climate against illegal immigrants, it is more important then ever that employers comply with the law with exactness. Failure for an employer to follow the law can result in fines and even jail for wilful violation. It is a serious matter, so don't be surprised if your employment application is rejected for failure to provide sufficient documentation of legal status to satisfy I-9 reporting requirements.