Various browsers
There are lots of different browsers out there. The most well-known are probably the following four:
The first one is Microsoft Edge, which comes with every Windows computer.
Other popular options include Google Chrome, Firefox and, if you're using a Mac (or iPhone), Safari.
Below is an explanation of these four major browsers.
Microsoft Edge

This used to be Internet Explorer back in the day, but Microsoft changed over. I don't recommend it, but people still use it. Believe it or not, it uses code from the same project its competitor Google Chrome uses, called Chromium.
Google Chrome

The browser that Google develops. Many people who don't want to keep using Edge will use this as their first choice. Chrome and browsers using the same code run off an engine called Blink.
Mozilla Firefox

A browser by the company Mozilla and the usual alternative to using Chrome. Firefox and other browsers based on its code are powered by an engine called Gecko.
Safari

Apple's own browser, only available for Apple devices. This uses an engine called Webkit, which Blink was originally based on.
Why should I care about browsers?
You can view any website using any browser. In fact, if you just want to get on with browsing the Internet, using Edge and not thinking about it is more than fine. But—and even though this is less important these days, it's still worth considering—there are differences in loading speed, startup time, usage of your computer's resources, and some additional features.
For a webmaster, the person behind a website, these differences are important, because some browsers like to display things differently to others. In the worst case scenario, which thankfully isn't common at all these days, a site that looks good in one browser will look awful in one using a different engine.
Back in the day, many tricks and a bit of abuse had to be utilised to get Internet Explorer to behave the same as other browsers. You can thank the lord that browsers are willing to act consistent nowadays!
Still, there can be differences in browsers' default states (especially with default line height), so it's important to make sure your website works in multiple browsers. Checking on at least both Firefox and Chrome when you make major layout revisions is enough. After all, you don't want people to visit your website and be unable to experience it!