While the U.S. Department of Labor's (DOL) Office of Disability Employment Policy (ODEP) does not enforce the ADA, it does offer publications and other technical assistance on the basic requirements of the law, including covered employers’ obligation to provide reasonable accommodations to qualified job applicants and employees with disabilities. For a quick overview of the ADA read “The Americans with Disabilities Act: A Brief Overview.”
Level AA is a little more significant, and makes sites accessible to people with a wider range of disabilities, including the most common barriers to use. It won't impact the look and feel of the site as much as Level AAA compliance, though it does include guidance on color contrast and error identification. Most businesses should be aiming for Level AA conformity, and it appears to reflect the level of accessibility the DOJ expects. 
Courts have taken essentially the following position on the issue of whether websites are places of public accommodation: Website accessibility fulfills the spirit of the ADA by lowering the barriers for people with disabilities to participate in business and commerce. As such, commercial websites need to comply with ADA regulations. Judges have reached this conclusion in several high-profile cases, such as the National Federation of the Blind’s lawsuit against the Scribd digital library. As a result of this case, Scribd agreed to redesign its website to work with screen reader software by the end of 2017.
...the full and equal enjoyment of the goods, services, facilities, or accommodations of any place of "public accommodation" by any person who owns, leases, or operates a place of public accommodation. Public accommodations include most places of lodging (such as inns and hotels), recreation, transportation, education, and dining, along with stores, care providers, and places of public displays.

Talk to your web designer about other techniques that will make your site more user-friendly for people with disabilities. Worried that’s not in your budget? Consider the fact that DOJ fines start at $75,000. And it's still yet to be determined if a non-compliant website is liable for one fine or will be charge per page for each violation. As the recent lawsuits illustrate, though, settlements quickly add up into the millions.