Browser Compatibility Issues
The best resource for seeing differences between browsers is, quirksmode, Compatibility Master Table It tries to document the differences between browsers for UI designers.
There are many different things that can cause browser incompatibility.. but here are some general categories.
CSS - Prior to IE10, not every CSS3 specification was supported in IE. This causes issues with layout and look and feel.
Rendering - Older browsers, like IE8, Firefox 4, etc.. do not support rounded corners. There are a number of different properties that are not supported, or have different names for. Sometimes in CSS code, you might see different properties assigned to target different browsers, for example gradients, (below is css to set the background color in a way that it supports Safari, IE8-10, Firefox, and Chrome.
Notice that there's 'webkit' settings for Safari and Chrome, 'ms' for IE, and 'moz' for Firefox, than a 'linear-gradient' fallback which is a proposed standard, but no one is following yet.
background-image: linear-gradient(bottom, rgb(43,216,52) 20%, rgb(69,255,78) 60%);
background-image: -o-linear-gradient(bottom, rgb(43,216,52) 20%, rgb(69,255,78) 60%);
background-image: -moz-linear-gradient(bottom, rgb(43,216,52) 20%, rgb(69,255,78) 60%);
background-image: -webkit-linear-gradient(bottom, rgb(43,216,52) 20%, rgb(69,255,78) 60%);
background-image: -ms-linear-gradient(bottom, rgb(43,216,52) 20%, rgb(69,255,78) 60%);
Ajax - Fortunately most developers will use jQuery or another framework to abstract out xhttp requests. But how different browsers handle it is very different under the hood. But you still need to be careful about security restrictions as different browsers will deal with CORS issues very differently. There's also defaults of what each browser consider a request timeout, and how many concurrent open connections a browser may allow. (some browsers may only allow like 4 out going requests at a time, where another browser may allow up to 48)
For quick testing of cross-browser compatibility of a specific site, you can you services such as Testize - Cross-Browser Web testing for Desktops, Mobiles and Tablets - it will detect many of the CSS and JS issues. I don't remember any other free services at the moment, but I'm sure there are others.