A domain is the unique name that identifies an Internet site. Domain Names always have 2 or more parts, separated by dots.  Example:  allicat.com 

The part on the left is the most specific (allicat), and the part on the right is the most general (com). A given machine may have more than one Domain Name but a given Domain Name points to only one machine. Usually, all of the machines on a given Network will have the same thing as the right-hand portion of their Domain Names. It is also possible for a Domain Name to exist but not be connected to an actual machine. This is often done so that a group or business can have an Internet e-mail address without having to establish a real Internet site. In these cases, some real Internet machine must handle the mail on behalf of the listed Domain Name. 

Choosing a Domain Name

Your domain name is what sets you apart from everyone else.  Try to choose something that people will relate to you or your business, and which is easy to remember (and spell).  

What if you cannot get the domain name of your choice?  It really depends on how committed you are to that particular name. If you have an existing brand name that you're known for, you'll probably not want to ditch that name just because you couldn't get the domain name. After all, it took you a lot of time and money to establish that name. If so, you might simply want to try to buy over the domain name from the current owner. Check up the "whois" information for the domain, and contact that person listed to see if they're willing to sell it. You should be aware that they are likely to want to charge a higher fee than you'd normally get when buying new domains (assuming they want to sell it in the first place).

On the other hand, if you're just starting out, you might prefer the cheaper alternative of trying to obtain a domain name first, and then naming your website (or business) after the domain that you've acquired. So if you've acquired, say, the domain name "FirstServe.com", then your website and business might be named "First Serve" or "FirstServe.com". I know this seems a bit like putting the cart before the horse, but that's the reality if you don't want to lose out on the Internet.

Get a shorter name if you can, but not the the expense of having a MEANINGFUL domain name.  Again, choose something that will be easy to remember (and spell) and that your customers will associate with you.  Some site owners use hyphens in their names, but they can be easily forgotten and so I discourage using them.  But if you really want a particular name, and the only way to get it is with hyphens, then go for it.  

One common question I get from people who can't get the ".com" domain of their choice, but find the ".net", ".org," or ".biz" etc., or other country-specific top level domains (TLDs) available (like .de, .nu, .sg, etc). Should they try for these?

Typically, it is better to have a domain name of your choice "mydomainname" even if it has a TLD of ".net", ".org", ".biz", ".us", etc. than some obscure name no one will remember.   You may consider a country specific name, like "mydomainname.de" or "mydomainname.uk" or whatever, but if you get a country specific domain, people might think that your business only caters to that country.  Also, .org is usually associated with non-profit organizations, though there is no rule against anyone else using them.

There are many companies through which you can register your domain name.  I usually use GoDaddy.com - but there are others.  Some hosting companies will provide FREE domain name registration if you host your site with them.