Found myself in nightmare of Cross Browser Compatibility!

Cross Browser Testing sound like what is going to happen. Yup it is!
I attended an interview 3 days ago. Interviewer asked me a question at the end, do you test web application on different bowser?
I said yes I do. Thay asked then, How?
I got stuck for awhile. Suddenly I broke my silence and said in cross browser I look for design issues. Unfortunately, I got discarded because of that answer. Don't know what reqruiter wants exactly.
Came back home but I couldn't stop myself thinking about that question.
I thought alot about this and finally found something interesting - Actually, What I realized in cross browser are CSS and HTML issues.
CSS is the one who manage the whole layout of the HTML.

I got something interesting about cross browser on adove page are below:  

Three levels of potential browser-support problems can arise:

• An error indicates CSS code that might cause a serious visible problem in a particular browser, such as causing parts of a page to disappear. (Error is the default designation for browser support problems, so in some cases, code with an unknown effect is also marked as an error.)
• A warning indicates a piece of CSS code that isn’t supported in a particular browser, but that won’t cause any serious display problems.
• An informational message indicates code that isn’t supported in a particular browser, but that has no visible effect.

Browser compatibility checks do not alter your document in any way.