A domain name is the way that humans recognize Internet locations such as websites and email addresses. It must contains two or more parts, separated by dots. The last part is always a top level domain like.com, .net or .org. In some countries, a top-level domain may also have a country code, like .co.uk. The other parts of the domain name are chosen by the person or group who register them for their own use. Websites only have to use one domain name, but there are several reasons why a company or organization might choose to use multiple domain names.
Protection from Cybersquatters
Domain names are registered on a first-come-first-served basis. That means if a person or company has not registered a name they use as a domain name, somebody else can purchase the rights to it, and attempt to sell it back at a profit. This is known as cybersquatting. Cybersquatting doesn't always work, particularly when prior trademarks exist, but it can be a complex process to reclaim a domain name due to the international nature of the Internet (and resulting heavy competition). Usually it's simpler just to register any domain names you think you might need to use in the course of your business, before someone else does.
Branding
Some companies have a large number of brands they want to advertise. A shoe company, for example, might want to sell both business shoes and trainers. Rather than have one site for the entire shoe company, they can register two domain names, one for each brand, and use different marketing techniques for each site. Similarly, people and organizations can set up separate sites for sub-projects, such as competitions or educational initiatives--these will often have a different style to the parent site, so they are kept under a separate domain name.
Multiple Points of Entry
A website does not have to have only one domain name. You can have any number of different domain names pointing to the same website. This is useful, because a company might have different names in different countries, or might want companyname.co.uk to point to the same place as companyname.com. Sometimes the names of companies change, but oldcompanyname.com can still point to the newcompanyname.com website, so there's no lost sales from people who did not know about the change.
Spelling
Everyone mistypes domain names occasionally when entering them into a browser. Just as normal names can be registered, so can misspelled names. These can be very valuable for advertising purposes; some websites buy up domain names that are common misspellings of own name, to protect their image (and direct traffic to their site).
Regionalization
An international company might want to have specific sites dedicated to specific geographical groups of customers. A French customer won't want to see prices in US dollars, so a domain name like company.fr can point to a different website than company.com. This not only allows the company to better target specific groups of customers, but they can also make sure their websites don't infringe on any local laws of customs for the area.