In the process of domain name registration you lease a domain name for a specified period. For this time you are the "owner" of domain and can use it for your own needs. In the meantime, this domain can not be used by anyone else and can not be registered, because each domain is unique and can not be repeated.
The registration process is very easy:
Enter desired domain name into verification form for free domains and click on "Check domain". If the domain that you have checked is still available, the wizard will let you know and you will be able to proceed with the registration. Once you have completed the process, you will receive an invoice within next few minutes. The domain will be registered once the invoice is payd.
In case when the wizard reports that domain is already taken, this domain can not be registered and you need to find another name.