Your customers could be browsing your website from many different devices and using a selection of web browsers. The most common web browsers are: Internet Explorer from Microsoft; Chrome from Google; Safari from Apple; and Firefox and Opera. So your website needs to be tested and work on all the different web browsers.
This, thankfully, is easier than it used to be. It used to be very difficult. When browsers were first starting out they had a nasty habit of interpreting the underlying code, the HTML, differently to each other. There were standards but how they were followed and understood varied from browser to browser. The result of this was that your website would look wonderful in one browser and all out of alignment in the next, and different again in another.
More recently, though, the web browser companies are converging and following the standards more closely, there are still differences, which need to be checked, but life for the web designer is getting easier on that front thank goodness.