If you want to have more control over your site, you will want to use a Content Management System (CMS). Compared to the first method, where each page becomes a separate HTML file, a CMS-powered site will store each page into a database. This way, all of your content is stored in rows and columns, which makes it much easier to manage. For example, you could display the latest 5 articles on your home page. If your site is built using a database, you can tell the home page to look inside the database for the most recent 5 articles, instead of creating all of the links by hand in HTML. A real time saver!
There are two basic types of CMS software. One is a commercial license, where a software company charges a setup fee, plus maintenance, for using their software. This Enterprise-level software typically costs a few thousand dollars for the setup fee, and a couple hundred a month for maintenance. Examples of commercially licensed CMS software include Ektron and Red Dot.
The other type of CMS software is Open Source. Open Source software is free to use, and free to customize. It is supported by an online community of developers - thousands of nerds from all over the world (like me) develop, support, maintain, and upgrade the software so that they can use it for their own purposes. Frequently, a client who needs an open-source plugin will fund the initial project, and makes the resulting software freely available. Some developers also build commercial plugins which can be purchased, typically for less than $100. Examples of such Open Source CMS packages include Joomla, WordPress, Drupal, and OSCommerce.
Also, a CMS-powered site will be easier to enhance, upgrade, or re-skin in the future. Many CMS packages will have basic functions like search, rss feeds, and image uploading capabilities built-in. Also, a strong CMS platform (my personal favorites are wordpress and Joomla) will have a plugin system which will allow you to add features to your site such as eCommerce, photo galleries, client extranets, online communities, discussion forums, and more.