To download YouTube videos on your computer...
You have a few different options when it comes to downloading to a desktop, starting with special (free) software. One of the more popular and reliable in this realm is called 4K Video Downloader, which is available for both Mac and Windows and makes it easy for you to grab hard copies of any YouTube clips in up to 4K quality. Once installed on your machine, launch the application, drop in your video-of-interest's URL, and voila, it'll automatically save to your downloads folder.
If you'd prefer not to add another piece of software to your computer, there are also a multitude of YouTube video "grabbers," which facilitate the conversion of videos into certain file formats and provide a simple a download link. These sites will also often allow you to convert YouTube videos to MP3 files, if you just want to save the audio.
There are literally hundreds of these types of sites out there -- and many disappear due to legal threats. Search and you'll find the steadiest, most professional sites around. One important note: Some sites can look and feel a little sketchy, so if it seems like you should steer clear of one for any reason (or your browser warns you as much), you probably should.