One thing you need to know is that domain registration is actually available for the public and people who are interested in this kind of service through the registrar. The costs and services you have to pay and that you might get vary greatly from one company to another, and this is very important. But usually, don't expect to pay a lot for this kind of service as it's relatively affordable.
Now another thing to do is that the domain name must be approved, and this can take longer than expected. What happens is that new names will often be examined and compared to existing names in the DNS database. Online services or the best domain registrar will often give you fields that you can fill, including the hierarchy that you are interested in, and from there, you will be able to find out whether the name you have chosen or not.
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During the process, it might be interesting for you to know that this is the point where you can then fill in your own contact details, which will make the whole process public in the first place. Of course, the WHOIS machine and the database will be some of the precursors for this from the start. Make sure you own a domain name because some registrants maintain control over most domains that are being registered.
Also, look for hidden fees that might be favored here or there, and this can be very annoying for those who know that they pay more than they should. All in all, the domain registration process must be easy and pain-free, and usually, you will be able to get search results quickly enough. All in all, this is what you need to know when thinking about the domain registration mechanism.