If you're having trouble finding a domain name (whether because of crowding or cyber-squatters), check for a help facility on each registrar's site. Domain registrars typically house search engines that return a listing of available names similar to the one you want. When you search for a domain name at Namecheap, for example, you get both the status of that name and a list of suffixes available for that name. Maybe [Sitename].com isn't available, but [Sitename].biz or .org is.

Registrars offer a wide variety of registration durations—one year, three, five, and even ten. Be careful about registering for more than a year, though. First, there might be restrictions on your ability to transfer the domain name should the registrar give poor service. Second, the registrar could go out of business, leaving your domain name without a host. Check the policies closely.
Name servers. Most registrars provide two or more name servers as part of the registration service. However, a registrant may specify its own authoritative name servers to host a domain's resource records. The registrar's policies govern the number of servers and the type of server information required. Some providers require a hostname and the corresponding IP address or just the hostname, which must be resolvable either in the new domain, or exist elsewhere. Based on traditional requirements (RFC 1034), typically a minimum of two servers is required.
There are organizations that are called registries that manage each TLD option. When you register domain names, you would first visit a domain registrar who would in turn check with the registry that manages that TLD to see if the domain is available. If the domain for the TLD you want discovers that it is in fact available, the registry adds it to their registry and gives you access to it. If the domain has already been registered by another user for that TLD, you will be shown during the domain registration process that the domain isn't available.
The way that competition is supposed to work between registries, is that lots of companies go to ICANN and say “We think we can reliably manage .com, and we think it would cost us [$5] per domain.” ICANN reviews these bids and awards the .com contract, which lasts for a fixed period of time, to whomever they deem best-suited to the job. In this time, other companies can think of ways to manage .com better/cheaper, and try and win the next contract. This, in theory, would provide an incentive for registries to innovate, a big part of which would be in lowering costs.
For the most part, you can pick whichever TLD you prefer, though a few have restrictions -- you can't buy a .edu domain unless you run an accredited post-secondary school. Many, but not all, country-specific domains (.us, .uk, .jp) are restricted to citizens of the country in question. Unrestricted country domains are often used for "domain hacks," where the extension becomes part of the title, such as YouTube's alternate address, "youtu.be."
Domain names are often seen in analogy to real estate in that domain names are foundations on which a website can be built, and the highest quality domain names, like sought-after real estate, tend to carry significant value, usually due to their online brand-building potential, use in advertising, search engine optimization, and many other criteria.

Under no circumstances should you pay more to transfer a name than to get a new one. Check what the transfer will require. Does the new service handle the task completely? Or do you have to go into your current registrar's site and change the technical details manually? Finally, check the transfer policy of the registrar before registering your domain name.
The hierarchy of domains descends from the right to the left label in the name; each label to the left specifies a subdivision, or subdomain of the domain to the right. For example: the label example specifies a node example.com as a subdomain of the com domain, and www is a label to create www.example.com, a subdomain of example.com. Each label may contain from 1 to 63 octets. The empty label is reserved for the root node and when fully qualified is expressed as the empty label terminated by a dot. The full domain name may not exceed a total length of 253 ASCII characters in its textual representation.[6] Thus, when using a single character per label, the limit is 127 levels: 127 characters plus 126 dots have a total length of 253. In practice, some domain registries may have shorter limits.

Domain names are often simply referred to as domains and domain name registrants are frequently referred to as domain owners, although domain name registration with a registrar does not confer any legal ownership of the domain name, only an exclusive right of use for a particular duration of time. The use of domain names in commerce may subject them to trademark law.

If you do not have a web host, you can always allow the registrar to park your domain name at a temporary website specially set up for you. This way you can quickly secure your domain name before it's too late and still take your time to set up the other aspects of your site. As far as I can tell, most registrars (or probably all) automatically park your domain by default whether you ask them to or not, so if this is your situation, you probably don't have to do anything special to get it done. Some of those registrars also provide you with a free email address at your own domain name while it is parked at their site, like sales@example.com (where "example.com" is your domain).
There are numerous domain name registrars. Listed below are just a few, along with my comments, if I know anything about them. Note that the domain name industry is highly competitive, with prices wildly fluctuating throughout the year, every year, so it's impossible to really mention accurate prices below unless I spend all my time updating this page. Please check their sites for the latest rates. All prices below are in US dollars.
When you buy a Premium Domain name, you are also buying strong branding potential, high recall, and the ability to attract more traffic to your site. Finding a relevant domain name to your business will provide you a storefront online, allowing your customers the most access to your products as possible. Your domain name is an investment that is easy to map back to success, and the perfect available domain name is just around the corner.
GoDaddy — This extremely popular registrar (probably the biggest today) offers .com domain names for $9.99 (plus 20 cents) per year (or $6.99 plus 20 cents if you transfer from another registrar). They have a web interface to manage your domains, free web redirection (where people who visit your domain will get transferred to another web address of your choice), free starter web page, free parked page or free "for sale" page, and an optional private domain registration where your domain is registered in the name of a proxy company. Besides .com, they also sell .us, .biz, .info, .net, .org, .ws, .name, .tv, .co.uk, .me.uk and .org.uk, etc. Note that (as with all registrars) the exact price varies depending on which domain you are registering (for example some domain suffixes are more expensive than others). Both credit card and PayPal payments are accepted.
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),[4][5] 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.