If you already have a web host, obtain from them the names of their primary and secondary nameservers. Don't worry if you don't understand what these things mean. Just save the information somewhere. The information can usually be obtained from their FAQs or other documentation on their site, usually under a category like "domain name" or "DNS" or "domain name transfer" and the like. If you can't find it, email them. You'll need the information to point your domain name to your website after you buy your domain. Having said that, if you don't have a web host yet, don't worry. Just read on.
1&1 Internet — This is primarily a large web host that is also a domain name registrar. Like all registrars, there are different charges for different domain suffixes. For example, at the time I last checked, you pay $0.99 for a ".com" on your first year, then $14.99 per year thereafter. The fee includes private domain registration, which means that your particulars are hidden from public view (done by registering the domain in the name of a proxy company). You also get a free email account and unlimited email forwarding, DNS management, a free SSL certificate for your domain, etc. Both credit card and PayPal payments are accepted by this registrar.
The practice of using a simple memorable abstraction of a host's numerical address on a computer network dates back to the ARPANET era, before the advent of today's commercial Internet. In the early network, each computer on the network retrieved the hosts file (host.txt) from a computer at SRI (now SRI International), which mapped computer host names to numerical addresses. The rapid growth of the network made it impossible to maintain a centrally organized hostname registry and in 1983 the Domain Name System was introduced on the ARPANET and published by the Internet Engineering Task Force as RFC 882 and RFC 883.
The right to use a domain name is delegated by domain name registrars, which are accredited by the Internet Corporation for Assigned Names and Numbers (ICANN), the organization charged with overseeing the name and number systems of the Internet. In addition to ICANN, each top-level domain (TLD) is maintained and serviced technically by an administrative organization operating a registry. A registry is responsible for maintaining the database of names registered within the TLD it administers. The registry receives registration information from each domain name registrar authorized to assign names in the corresponding TLD and publishes the information using a special service, the WHOIS protocol.