The practical effect is recognition rather than persuasion. Somebody with several tabs open finds yours by the icon rather than by reading truncated titles, and a bookmarked site without one sits in a list of identical blank documents. Neither is decisive and both are the kind of small friction that accumulates into an impression of a site nobody finished.
Use the logo mark rather than the full logo, which is where most attempts fail. At sixteen or thirty two pixels a wordmark becomes an illegible smudge, and anything with fine detail disappears entirely. If your logo has a symbol, use the symbol alone. If it does not, a single letter in your brand colour is a perfectly respectable favicon and considerably better than a compressed version of the whole thing.
Test it at actual size before deciding, on a light and a dark browser theme. What looks distinctive at a hundred pixels frequently resolves into a dark blob at sixteen, and a design with a white background disappears against a light tab bar. This takes one minute and catches nearly every problem.
Supply more than one file, because the icon is used at several sizes across several contexts. Browsers want small versions, mobile devices want a larger one for the home screen, and some platforms want a specific format. A favicon generator will produce the full set from a single square image and give you the markup to include, which turns a fiddly task into a five minute one.
Include the home screen icon specifically, since that is the one people see if they add your site to a phone. A missing one produces a screenshot of the page as the icon, which is unreadable and looks accidental. For any business whose customers might return regularly, that is worth the extra file.
Check that it actually appears after uploading, because browsers cache favicons aggressively and a stale one persists long after the file changed. Testing in a private window or a browser you do not normally use gives you the visitor's view rather than your own cached version.
Match it to what appears elsewhere. The same mark should be your profile picture on your business listing and your social accounts, because those are the places people encounter you at small size and consistency across them is what produces recognition. A different image in each location is four impressions of four things.
It is a fifteen minute task with a small return, which places it correctly: worth doing when you build the site, worth adding if it is missing, and not worth a project. The reason to mention it at all is that its absence is one of a handful of details that make a competent site look unfinished, and unlike most such details it has a definite end point.
Check how your site appears when shared in a message or on social, since the same neglect that produces a missing favicon usually produces a missing preview image. Both are small files specified in the page head, both take the same few minutes, and both determine what your business looks like in the places people encounter it outside a browser tab.
Use a square source image at a decent size, since generators produce every variant from one file and starting with something rectangular means cropping decisions get made for you.