When I develop a site, I make sure it is able to display correctly on the latest systems and browsers (going back a few years at the point of developing).
If you want a mobile-friendly site, I make sure your site displays correctly on all of the most popular devices. Even if someone were to access your site from a really old mobile device or an obscure device (such as one of the many tablets in the market which has come and gone), your site content will still display but probably not as well on newer or more popular devices.
If you do not want a mobile-friendly site (which is not recommended these days), your site will still display and visitors can still go from page to page, it just won’t be as easy to navigate or see it well if the site was not optimized for mobile devices.
One big reason to make sure you have a mobile-friendly site (aka responsive) is that Google and other search engines are now giving higher ranking in the search results if your site is optimized for these devices.