Obviously, the first question is this: which browsers are popular among your users?

And you can easily find out which browsers, operating systems, and devices are popular among your users by using Google Analytics.

Go to Audience -> Technology -> Browser & OS:

As you can see, here you can find out information about browsers, operating systems and screen sizes used by your target audience.

OK. Now that you already know which are the most important browsers for you, let's see the available tools.

1. Browsershots

http://browsershots.org/

Browsershots will allow you to test your website with several browsers on different operating systems.

It's free, but it doesn't include Microsoft Edge or Internet Explorer or mobile device support.

All you have to do, is to enter your website URL and then select the browsers you want to test.

Browsers are listed in three columns for Linux, Windows, and Mac OS.

After submitting your request, you will have to wait a bit while the application runs the tests and generates the screenshots.

Once finished, you’ll be able to see all the screenshots on the same page. Just click on a screenshot to expand it.

And you need to keep in mind that we are talking about screenshots, and NOT live-testing.

In other words, you won't be able to see if your site works as intended during the tests. For that, you’ll have to run live tests in actual browsers.

2. MobileTestMe

http://mobiletest.me/

Once you log in you'll be able to select the desired device. Then just enter your domain URL and click the little blue arrow to generate the according preview.

And of course, there are many other free and paid tools and browser addons out there. Just make some research and you'll find your favorite solution.