Domain names exist because for humans it is much easier to deal with names like google.com instead of IP addresses. DNS servers translate domain names into IP addresses that are needed to actually transfer data.
A complete domain name like ip-checker.info consists of name part (ip-checker) and a so called top level domain (info, com, org, de, ...). Each top level domain is administered by an organization called registry, where everyone can register a domain name of his choice on a first come first serve basis. The US based ICANN organization oversees and coordinates all these registries. A domain name may be extended with subdomains or host names like www. The complete domain name would then be www.ip-checker.info.
If you register a domain name you own it like a landownwer owns a property. Similar to a land registry the top level domain registry keeps a record of your personal ownership and everyone with a valid interest may request information about your ownership. Unfortunately all personal information is nowadays hidden from public access due to privacy regulations, but at least you will get the name and contact data of the internet provider where it is possible to request disclosure of the hidden personal data.
In order to translate a domain name into an IP address the internet has a hierarchical, decentralized Domain Name System. This DNS System plays an essential role in the internet. There are millions of DNS servers around the world that are authorative for only a small subsset of all domains, but know which name severs to contact for all other domains. This way no DNS server needs to keep a copy of all domain names worldwide.