Welcome to DNN! Your existing site appears to me to be what I thought of before DNN as a 'normal' (and very artistic) html-based web site. DNN is in many ways a radical flash forward in web technology, a completely re-thought way of presenting material on the internet. The heart of the difference is DNN's use of a MS-SQL database to store your information for retrieval and display, and DNN's exploitation of server resources to allow you to host many web sites or web portals off of one installation of DNN.
There is a free forum module that can be downloaded from the DNN project site, and there are several for-pay forum modules you can purchase at Snowcovered.
My perception of the development of web site technology is that after the html coding approach I think I see in your existing site, there came, in several different programming/scripting languages, a database-driven approach where a visitor to your site gets a page essentially generated on demand in ASP, PHP, or another language. There is a free, highly-developed forum package (Web Wiz) written ASP, for example, that can be dropped into any one's existing HTML site or woven into an ASP-based portal package. DNN, using ASP.NET, extends the power of this approach and allows one installation to serve/manage many portals. It can also serve/manage just one if that suits you.
To simplify my message: Web Wiz might be what you want if all you want is a free state of the art forum added to what you've got. If you want to use DNN, you would backup your existing material, install DNN, possibly in its own folders on your web server (or have your web server admin do it), and then set up a DNN portal and migrate your material over to it. If you want a forum in this scenario, you would add that module to your DNN installation and then set up what you need on your new DNN portal for your material.
Getting your existing "look" might be a bit of challenge. The DNN package is relatively feature-rich and style-poor. Your artisitic talents would be very valuable around here if you were able to apply them to extend the style options of DNN!