In short; Yes there are big computers that store all that information. Usually your Internet provider has one (or a few) that knows all IP numbers and IP names of all systems and customers they have, and of most large 'number blocks' that are located elsewhere. Also there are huge central machines that know where to find more detailed information about all number blocks.
Main point; your PC automagically uses the DNS server(s) that your Internet provider has available and there is nothing you should worry about. If it works it works, and when it is broken there is nothing you can do to fix it.