Favicons are small square 16x16 icon files displayed next to the URL of your site in a browser's address bar.
These small iconic images show a graphical representation of your website, often displayed next to the name of your site in a user's list of open tabs and bookmark listings, making it easier for the user to easily identify amongst other sites.
It was Microsoft's Internet Explorer 5 released in March 1999 that first supported this file. It was denoted as 'shortcut icon' in the rel element of the code that was placed in the section a site. In December 1999, the favicon was standardized by the World Wide Web Consortium (W3C) with the recommendation of Hypertext Markup Language (HTML). Read further for more on the history of Favicons.
If you have bookmarked a website or added them to your home screen, it is the favicon that makes them identifiable as the websites that they are. So that takes care of the what are favicons bit. Now why are they important? They are important for the exact same reason that any other branding activity is important.
Establishing brand recognition is very important for SEO, not just digital marketing in general. If you are doing SEO for your company, a website's favicon will help searchers recognize that website in the search results.
If your site doesn't have a favicon a web browser will display the default, blank document, on a browser tab. Worse still, if this is displayed next to a competitor site then you are going to lose credibility and trust.
Apart from branding, favicons are also important for a good user experience. Imagine if you have a 100 pages bookmarked, scrolling and reading through them is harder than simply looking out for the favicon and clicking on that page. And everyone knows that a visual is far more recognisable than text. So this is definitely better for user experience.
It also helps users understand that they are indeed on the right website. After all typos are common and landing up on a website with a familiar favicon can help reinforce trust.