It's quite common for a site to be referenced by more than one domain name. In fact, most sites are referenced by at least two: a www version and a non-www version. These are usually set up to reference index page on a site and produce same results for a searcher.
It could, however, be argued that these are these same domain names. So question remains: why would someone want to have more than one unique domain name for a single site?
Search engines - First, let's take a brief look at search engines. In past, it was a very common spamming technique to purchase dozens, hundreds or in some really gross cases, thousands of domain names, all referencing exactly same site. These were all submitted to search engines, and many of them were indexed and blindly added to results. This is how many questionable sites used to get top search results very quickly and inexpensively.
The search engines have apparently caught onto this technique. At very least, it has become common knowledge that this kind of spamming is not tolerated (sometimes common knowledge can be just as effective a deterrent as actual enforcement). I know that in past it was normal to find many sites of different domain names but identical content in search engine results; today it's far more rare.
In fact, top search engine, Google, bases it's ranking scheme on quality of links. What this translates to is you must get popular (higher ranking) sites to link to your site to raise your ranking. Thus, it's a better strategy to get as many links to a SINGLE domain name than to many different domain names.
With this in mind, it's now considered best by most search engine optimization specialists (at least those that know what they are doing) to only list a single domain with search engines, perhaps with www and non-www version but nothing else.
Multiple entry points - One technique that I use on my own site with great success is to have multiple entry points, each it's own domain name. Let's consider a mythical site in order to illustrate how this works.
The site is about homemaking, and thus main domain is "homemaking.com". Underneath this are sections about sewing, housecleaning and cooking. You might use "homemaking.com" for link exchanges and search engine submissions, then create three additional domains: "sewing.com", "housecleaning.com" and "cooking.com" (although if you actually managed to purchase those domain names you could resell them for quite a chunk of change).
Each of these domains would use a 301 redirect (this informs any search engine that page has permanently moved to a new location) to a specific page on site.
Those three domains would then be used in different themed marketing campaigns. You might submit an article to a cooking site, for instance, which referencing cooking.com. For a newsletter about cleaning, you would use housecleaning.com. Each domain name is merely a shortcut to master domain, but it is much more targeted than "homemaking.com".
Protection - If you own a business, it's a great idea to think of some of derivations of your site name and purchase those as well. Thus, if you had a company named "xyz", you might also purchase "xyzsucks" and "ihatexyz" as well. You may as well direct these to your site, but be sure to include 301 redirects, as you definitely do not want them in search engine indices.
Typos - Sometimes people misspell things, and domain names are no exception. Knowing this, you can get some respectable traffic by purchasing common misspellings for your domain name. Just remember to use 301 redirect method so these misspellings are not listed in search engines.
Other TLD's (Top Level Domains) - If possible, it's a good idea to get .com, .net and .org version of your domain at a minimum. I tend to get .us (or whatever country is appropriate), .info and .biz versions as well. This ensures that no matter what people type they will get to your domain. Of course, remember to 301 redirect these domains so they don't get listed.
For branding purposes, it's essential to get other TLD's if you can. If you don't you may be embarrassed to find some pornographic or casino site has purchased your name with a different TLD. The white house site (whitehouse.gov) is a classic example: .com version has nothing to do with white house (if you type this URL, be sure your kids are not present).
Other TLD's with different content - In a slight alteration of above method, I have purchased additional TLD's, but made each one slightly different. To use above housecleaning example, housecleaning.com might be a page about housecleaning in general, housecleaning.us might index articles specific to United States, and housecleaning.biz may include information related to housecleaning businesses. Each of these is just a page or two, and links back to main housecleaning.com domain.