All browsers differ in the way they interpret and render code, so it's important that you test your site in at least one version of each of the major ones: Chrome/Chromium, Firefox, Safari, Opera and Internet Explorer. The first four are very similar, but still, a small coding error that is disregarded in one could break down in another.
The big exception and troublemaker is Internet Explorer, which often goes its own way. Globally, Internet Explorer is losing ground, and particularly the more idiosyncratic versions IE6 and IE7 represent barely a few percent of all users: Explorer statistics
(IE6 is not even supported any more by Microsoft, and IE7 was a failed version which will probably go the same way soon)
Use your host's CPanel statistics for visitor's browser and OS usage (AwStats is common and good enough), or integrate Google Analytics into your website. After a while, this will give you a more accurate picture of what browser versions, OS and screen resolutions should be your main focus