Mozilla websites should be tested in a broad range of browsers, though that doesn’t mean a site must look or behave exactly the same in all browsers. Sites should look good and work well in any modern browser, even if it’s not perfectly identical.
The current generation of browsers (versions released within the last few years) are all pretty even in their support of most modern web standards, but many people still use older browsers and deserve equal access to information. Consider following a system of graded browser support wherein the oldest or least capable browsers can still access all the content and essential functionality, just without all the bells and whistles better browsers can enjoy. If you use a progressive enhancement methodology, support for older/other browsers may already be a given.
Unless there are specific reasons not to, you should look to fully support (at the very least) the latest versions of all major browsers (Firefox, Chrome, Safari, Opera & Edge), as well as the last 2 versions of Internet Explorer (10 & 11). All other browsers can have degraded support.
Don’t use proprietary features supported only by a single browser unless what you’re making is a demo of that specific feature. Be mindful when you use emerging standards supported in some browsers but not others and include fallbacks for less capable browsers. Check MDN or CanIUse for compatibility info. If you use vendor prefixes in CSS, include the prefixes for all supporting browsers as well as the unprefixed standard.
Finally, don’t rely on user agent sniffing to determine which browsers can and can’t access a website or use its features. Instead, favor feature detection that allows older browsers to degrade gracefully in what they show.