A browser is the program you use to open and look at websites — well-known ones include Chrome, Safari, Firefox and Edge. Every time you read a page on the internet, you are doing it through a browser.
Think of it as a window onto the internet: the website itself lives far away on another computer, and the browser is the glass through which you see it and interact with it — clicking links, filling forms, watching videos. The same website can be opened in any browser, and a well-built site should look and work correctly in all of them.
When you type a Domain into the browser and press enter, the browser goes and fetches that website and shows it to you on your screen.