The best website in the world doesn't do anybody any good if they can't find it. The ability to find a site—or anything else on the Internet—depends on the Domain Name System and its capability of taking human-readable addresses like "example.com" and translating them into addresses that computers can understand like "192.0.32.10". Most web sites have a fixed address; Google will always be at google.com, Twitter is at twitter.com, and so forth. So what about web sites that don't have a fixed address?
Suppose you want to host a site at home or at work. Now suppose you don't have a fixed address. Maybe you're on a dial-up connection—in which case you should seriously think about not hosting—or more likely you are on a broadband connection that doesn't include a fixed IP address. How will anybody find you if you're never in the same place twice?
DynDNS is a service for situations like that. You install a little app that lets their servers know what your current address is, and when people want to visit your site, they are directed through some back-end sleight-of-hand to your site, even though it's not located in the same place today that it was yesterday. They've got an update client for both Windows and Mac servers, and give some instructions for how to configure a Linux server as well. So now you can host a server at home (subject to your ISP's terms of service) and people can actually find you.