Question: What are the common tools in use to test website accessibility, i.e., what are the actions to take for each of the P.O.U.R. categories?
Answer: Onions have layers...
The answer is not a simple, use this tool or technique for each guideline category or even for each sub-category. Each sub-category can have several scenarios with 5 to 12 different techniques to employ depending on how the situation fits your website.
Example:
Perceivable - 9 sub-categories, Guideline 1.1.1 Non-text Content - Level A
The W3.org resource provides six scenarios (A through F) with detailed resources on how to meet and when the guideline does not apply. There are also advisory techniques provided.
There are two resources I would recommend to learn about advisory techniques and failures of each of the guidelines by sub-category and level (A; AA; AAA).
The first is the W3.org, WAI, WCAG Quick Reference. This resource provides a complete description of the guideline, success techniques, and failure examples.
The second covers the new 2.1 guidelines added this year. This resource also provides complete descriptions along with how to understand each and how to meet each.
Keep in mind, you are designing for several audiences which includes users of digital and hardware assistive technology. Proper use of semantic HTML, CSS, and WAI-ARIA is essential.
Important: always use WAI-ARIA in addition to HTML,
not in place of.
In General, common tools/methods used include:
Design/Development: Proper use of semantic HTML and CSS along with WAI-ARIA to provide additional utility to assistive technology. Remember, WAI-ARIA is used in addition to HTML and does not replace HTML.
There are several plugins you can use to test as you code:
DEQUE aXe, which includes a link to DEQUE resources providing an explanation of the issue, why it is important to fix, what code the issue is associated with, how to fix the issue with a tie to the WCAG and/or Section 508 guideline(s).
WAVE, also provides detailed explanations of issues and how to fix along with what code the issue is associated with.
Accessibility Inspector, Firefox also has an Inspector feature in the Developer tools that can be set to include Accessibility checking, reference, MDN web docs, Accessibility inspector.
QA Testing: Manual testing according to DHS Trusted Tester methodology which includes keyboard only; gesture testing; zoom scaling; auditory/vocal; and screen reader testing along with automated scan tools such as Sort Site by Powermapper; Total Validator; and Tenon to list only a few.
Keep in mind there is no substitute for design-time inclusion, reference, an article on Smashing Magazine, Designing For Accessibility And Inclusion by Steven Lambert.