There are two approaches to accessibility testing: manual and automated.  Accessibility test tools make automated testing possible and manual testing much easier.  Most tools are free, but some, especially the fully automated test tools, have a cost.  There is no one test tool that can check everything so you will want to build a suite of tools to use when testing the accessibility of your website.

WebAIM provides a good breakdown of considerations when choosing between Accessibility Evaluation Tools.

Novice-friendly Test Tools

These novice-friendly manual test tools will take you a long way in accessibility testing with only a small amount of knowledge or training needed.

Expert-level Test Tools

These manual test tools are intended for use by developers or expert accessibility testers. They all provide the testers with the ability to run tests against a single web page and then drill down into the code to determine what is causing the error and correct it.  Like the fully automated test tools, these tools test around 30% of the required criteria for accessibility; the difference is that tests must be run manually one page at a time.  Each of these free manual tools can be used in conjunction with a fully automated tool from the same company.

Enterprise-level Test Tools

Once configured, these tools will automatically test, monitor, and report on the accessibility of an entire site or sites at scheduled intervals.  The in-depth reports generated by these tools provide novice testers and accessibility managers with wide-ranging insights for managing accessibility errors over time, while still allowing expert testers access to information needed to start troubleshooting errors. This expanded functionality has a cost, but the ease of use offered by this solution may well be worth the price.  Contact the vendor for the pricing for all these tools.

Simulators and Assistive Technology for Testing

Simulators and assistive technology such as screenereaders let you experience firsthand what people with disabilities experience when they visit your website.  They are an invaluable part of your accessibility test suite.  WebAIM provides basic tutorials to get you started with screenreader testing for JAWSNVDA, and VoiceOver.