Static Sites
These are sites that are like an online brochure. They aren't intended to change very often and they don't interact with the visitor (allow them to subscribe, or leave comments or log on), they just display information.
They would have information on different aspects of your business and some photos most likely, and your contact information so people could reach you through your site. If you were updating it with a new page or something, you would most likely call your web designer to do it because there is no editing software attached to the website. These sites are just connected web pages.
If you don't need to update your site often this is the best way to go because the pages are efficiently light and load fast.
Content Management Systems (CMS)
These are sites that are designed on a user friendly platform that makes publishing new content quick and easy. They are used by companies that regularly make updates or add new content. A good example would be a media site, but a business that adds frequent new content such as press releases and pdf downloads would also manage better with one of these systems. They are updated with a visual editor that looks much like a word processor. An online shop site is also a CMS.
Ordinarily, a website is just designed to display information on a browser. A CMS is a little bit different in that it is actually a complete piece of software in disguise. To the person browsing the site all he or she sees is the web pages. But what the person doesn't see is the second side of the website, which is a whole website administration system. The owner of the website (or anyone with the correct authentication) can log into the admin area and have a whole other series of options presented to them.
This website is actually a CMS, and if you were logged in as the administrator you would see an additional link to the left that says "Administer." If you were to click on that link you would have a whole range of options presented to you in the main page. Options such as: list content, add pages, add FAQs, delete pages, edit pages, etc. These would be the options most users would be interesting in when updating their site. The more advanced user might be interested in other options like: add new menu block, change meta tags (the way search engines index your site), install additional modules (like a shop or forum), administer permissions (who can view what content on your site), etc. On this particular system you have a very fine level of control, on other simpler systems, not so much but still enough to make changes to your own site.
The benefits of using a CMS are, of course, that you don't need to call your website designer when you want to make minor changes to your site. This is a huge benefit if your website designer charges $40 per hour and you're the sort of person who changes your mind often.
The level of know-how required to use most CMS's is not advanced. It does take a bit of time to get orientated to all the new options and what they do, but generally if you have been able to learn how to use something like a modern word processor, you should also be able to learn how to use a CMS.