How to register a domain name
Go to one of the many registars, such as Register.com, Network Solutions, or Godaddy.com. Use their search engine to see if the domain name you're interested in is available. You can also use the "who is" function to look up who owns a particular domain name. If the name you want is not available, the registars will offer you other similar ones, usually those with different suffixes than .com, such as .net or .biz. These suffixes are called "top level domains." Once you settle on one, complete the registration process by entering your contact information and paying the nominal fee.
How NOT to register a domain name, part 1
Do not let someone else register your domain name in THEIR name. Even if you've hired them to design your website. Even if they're your partner --if they're your partner, register the partnership as the registrant, and make sure you have a written partnership agreement that proves the ownership. The reason this is important is that registrars ask for the registrant/owner's s name, an administrative contact, and a technical contact, and the registrar will NOT ask any questions about the registrant's internal organization or agreement, they'll just take the registrant's word for who is who.
How NOT to register a domain name, part 2
The registrant is the owner of the domain name, the administrative contact is the one who gets the yearly or bi-annual registration statement and should pay the required fee, and the technical contact is the computer/tech person who may be required on occasion to change technical settings and to troubleshoot technical problems. If you don't make sure that the actual owner of the domain name is listed as the registrant, you may end up in a dispute, and the registrar, who more or less functions as a utility and charges a nominal fee in exchange for marketing these domain names, has a contract with all registrants that limits their liability in case of a dispute over ownership.
Maintaining your domain name's registration
Your domain name, like trademark rights, need to be periodically renewed, and can be kept indefinitely. Your registrar will send your administrative contact renewal notices, and the required renewal fees need to be paid. Make sure your administrative contact's address stays updated, and make sure these fees get paid --otherwise your rights to your domain name can be lost. Ignore unsolicited notices you may receive from scam companies that may look like your registrar but aren't. While they may offer to renew your domain name, they charge a fee for this service, and you don't need to use them, you (or your administrative contact) can simply renew your registration when it's time to do so.
The value of a domain name
As everyone has realized that a short, recognizable internet address can garner great attention and reap huge profit, and because domain names are so cheap and easy to acquire, there has been a great deal of speculative buying among web entrepreneurs. This demand has led to a relative shortage of .com top level names, and soon new top level names will be become available. The reason for the speculation is clear: Domain names in and of themselves can be wildly lucrative. One of the early pioneers of sex on the web had his www.sex.com domain name stolen, and he was awarded $40M plus another $25M in punitive damages for its theft. See Kremen v. Cohen, 337 F.3d 1024 (9th Cir. 2003), which also found the registrar subject to a "conversion" claim for giving Kremen's domain name away as a result of a clumsy forged letter.