You can register a domain name for yourself by going to any of the domain name registrars worldwide. If the domain name is still available, you will be able to register a domain name for yourself under normal circumstances.
Every day thousands of domain names expire. Some owners no longer have any use for the domain name and let it expire. There are owners who have no idea that their name has expired. Once the domain name expires it can also be referred to as a deleted domain name.