As someone who develops a lot of websites, one of the biggest issues I come across is developing a website to look (almost) identical in all major browsers. The most popular browsers are Internet Explorer, FireFox, Chrome and Safari.
It is important to have a website look and function similarly in the different browsers that your visitors use. You want to provide your users with a consistent experience at all times.
How I tackle the cross browser issue:
BrowserShots.org - It can take a screen shot of how your website renders in all different browsers. Not my favorite tool to use and you only have an image to work with and cannot test functionality.
Install the major browsers on my computer so I can actually play around with the website. This may require running a virtual machine(s) so that you can have different versions of the same browser. For example, Internet Explorer 7 and 8.
Spoon.net - Allows you to run popular browsers on your computer without actually having to install the browser. Unfortunately, Microsoft does not allow Spoon.net to provide Internet Explorer.
Another good idea is to install some sort of analytics that will track which browser your visitors are using. This way you know to make sure you test your website in that browser for any issue that may arise when changing code around.
Finally it has been getting easier to develop websites and not having to worry about inconsistency issues across the browsers. Most browsers now follow the standards so you really only need to develop the site and it will look the same in all browsers.