Databases are used to store data. They can help a great deal when web sites hold a medium to large amount of information. Many web design companies use them to hold individual page information for their clients. They then design a template or skin or wrapper that calls the information as needed.
As an example, visit any popular news web site. You'll notice a general structure to the site that is its template or skin. You'll notice content in the form of text or images, etc. Typically content that swaps out from page to page on these large sites is stored in a database.
Look up at the address bar (where the URL is). You will notice numbers and words that don't make sense. Sometimes you see "id=45689" or something similar. That is the record where the information is stored in the database. The links on the template point to that record and that is how it is pulled up in the template.
Many website owners who run their own site wouldn't want to mess with learning about administering a database or developing around it. Your site will usually only have a database "backend" if you hire a company to design it. It is a great option if you have a company to do it for you.
Benefits of having a database and template web site
The key benefit is adding new pages and editing existing pages is relatively simple. Think of how difficult it is to add a new page to a blog. It takes seconds to add a post. Most blog tools rely on a database. Another benefit is found in page management. Designers can categorize pages in many ways and pull them up easily. Just think of how easy it is to link a blog entry to many community groups. Within seconds you can add your link all over the place, or remove it if you need.
Downsides to using a database
The major downside to using a database is that you have to know how to do it. It is not a simple technology for many people. Rarely you'll get someone who can just pick it up. You have to pay for great development usually. It can be costly. Another thing to keep in mind is to backup the database frequently. What happens if one of the database pieces becomes corrupt? You may loose your whole site.
Templates and databases can save a lot of time in the long term. Consider using Vauntium to build what you need with database templates and you may have a huge head start in your site's deployment.