Here's how you can register a domain name using a domain name registrar.
Visit a domain name registrars homepage (e.g. namecheap.com).
Usually there is a large input box/text field where you can enter the domain name you want to acquire. Otherwise you'll need to browse the site until you find a link leading to a page where you can search/purchase a domain name.
Enter the domain name you want to purchase and press enter/return or click on the search/submit button (if there is one).
You should see a list of domain names (based on your query) using various domain extensions. For example, if you typed compuglobalhypermeganet.com, you will see a list of domain names, your request should be at the top, followed by something like:
compuglobalhypermeganet.com compuglobalhypermeganet.net compuglobalhypermeganet.org compuglobalhypermeganet.ca compuglobalhypermeganet.us compuglobalhypermeganet.tv ... etc.
Some domain name extensions/country codes are only available to citizens of the related country. For example, only Canadians can register a .ca domain name. However, anyone can register a .ly domain name even though it is a country code for Libya. Any decent registrar will inform you of any citizenship or residency issues before you complete a purchase.
The list should provide some indication of which domain names are available and which names are unavailable. Usually unavailable domain names will have grey/faint text; whereas available domain names are usually in bold text accompanied by a shopping cart icon or a plus sign and/or a price.
If the domain you want is unavailable (or listed at a price you are unwilling to pay), you choose one of the alternative domain names provided or search for a different name until you find an available name that you desire.
Registrars often provide the ability to register multiple domains in one order. So you should be able to add multiple domain names to your shopping cart. Also, you should also be able to search for additional domain names and add those your shopping cart as well. When you are ready to complete your order, look for some kind of "checkout" button--usually it will be a shopping cart icon located at the rop-right corner of the page accompanied by the sub-total (and/or total number of items) of your order.
When you decide to checkout, you will be asked to provide contact information to the registrar along with contact information for ICANN, the non-profit organization that's responsible for all domain names. ICANN requires three sets of contact information: owner, administrative, billing, and technical.
The ultimate owner of a domain name who has the right to use, sell or destroy a domain name.
The person that works on behalf of the registrant and can do anything that the registrant can do.
The person responsible for paying the renewal fee.
The person responsible for dealing with technical matters related to the domain name.
Most domain names are owned and managed by one person, in which case, you only need to provide one set of information (name, address, email, phone number). Your registrar should provide an option to apply your contact information to all categories (registrant, administrator, billing, technical) automatically.
Providing contact information for these various roles does not grant those individuals or organizations access to your account on a domain name registrar's web site (e.g. namecheap.com). (However, your registrar may allow you to create sub-accounts for those individuals and others.)
It's important that you provide accurate information (including your name) in case there's some sort of legal dispute over who owns the domain name.
NOTE: I can't explain why ICANN needs the name and address of the person responsible for paying your renewal fees or your tech guy. I have scoured the web looking for an answer and I have been unable to find one.
All of this information is entered into public database called "WHOIS". A WHOIS database lists the contact information for all top-level Internet domain names.
Most registrars provide an option to hide your contact information and they accomplish this by listing their own mailing address and a randomly generated e-mail address (email@example.com) that forwards e-mails to the real email address (firstname.lastname@example.org) you have provided.
The WHOIS database allows people to contact you even if your contact information isn't listed on your web site. Most of the time people use WHOIS to send unsolicited emails promoting products and services. But there are also legitamate reasons for why a stranger might want to contact you. For example, someone may want to purchase your domain name; or maybe someone might want to discuss a legal matter such a copyright or trademark infringement. Also, ICANN will need to contact domain registrants from time to time.
It's important that you provide real contact information. If ICANN somehow determines that the information provided to them is inaccurate, they may revoke ownership of the relevant domain name(s).
The big reason to provide contact information for multiple inividuals is so that if you get hit by a bus, there would be other people on the record who could make a claim to get control of your domain name.