Critical areas for browser compatibility checks are missing functionalities due to lack of support for javascript in the specific browser. It should also take into account the GUI issues like misalignment.
Some of the browsers may not support Javascript or Active-X control which are required by the software to provide the desired functionality to the end-user.For See the following wikipedia article http://en.wikipedia.org/wiki/Comparison_of_web_browsers#JavaScript_support
For example:
1.) The date input field is validated for incorrect date-formats using javascript. If javascript is not supported by the browser, the software will allow the customer to enter invalid date which is a defect
2.) A country drop-down is sorted alphabetically using a client-side javascript. The same rule applies as one. If it is not supported, the user finds it very difficult to select a country from the dropdown
Text, graphics and color can appear differently on different browsers.
1.) Misalignment of controls on the web page which looks frustrating to the customer
2.) Web page looking cluttered on mobile browsers
