Free fantasy animated backgrounds can be a fun addition to your web page, chat window, or home video creation. However, there are some problems with creating a moving background in whatever medium you are decorating.
The Problem of Distraction
The biggest problem with any animated background, whether fantasy-based or anything else, is that when the background is in motion the human eye naturally travels to it and away from whatever other content you have on the page. In fact, the better the content, the harder it gets for the user, since they'll want to stay on your page longer. The more motion you put on the page (such as a repeating small GIF in the background) the more of a headache the page will be. This is why many web designers cringe at any constant motion on a web page.
This is not to say websites should not be dynamic - far from it. The same instinct that makes people distracted by motion also gives a visceral satisfaction when their actions cause motion. Using Flash to create interesting transitions or button effects, for example, is one way to have animations going in the background of your web site without interfering with content. You could even pull off a large, slow-moving background (such as a large moon traveling across the sky behind your site about werewolves) if you used Flash's programming language to move the object slowly enough that it registers without being distracting. However, motion and animation is a fine art, and in most cases shouldn't simply be thrown up into a site for the sake of having it there. An animation that looks great standing alone, such as this animated Matrix clip, would be disastrous on almost any web page.
Other Kinds of Free Fantasy Animated Backgrounds
Of course, there are other kinds of backgrounds - such as those for your desktop. However, for all the same reasons stated above, most fantasy backgrounds available for free online are static - even the ones that are on sites claiming they are animated. Unfortunately, having an animated Flash background can take up significant processor power on your computer, and most operating systems from Vista on don't support using animated Flash clips as the background.
However, the place that kind of background would work very well would be in a home video. Many basic video editing programs such as iMovie now support "green screen" effects - that is, having a character move against a solid color background and then replacing the solid color with some kind of other video - such as the Matrix effect, or a background like those available for free from Fantasy Art Design. Even the recent releases of several chat programs like iChat from Apple allow the background to be replaced by a video clip, as fantastical as you like.
Sources for Fantasy Backgrounds
There are a wide variety of sites offering background art, from 3-d figure modeling to old-school tiled gifs with unicorns and fairies. Bloggers may find sites like Fantasy Graphics useful, creating a simple sidebar background but leaving the majority of the page free for content.
Because fantasy is so popular, though, you do need to be aware that some sites are not really interested in offering fantasy wallpaper as much as drawing your eyes to their advertising - or worse, tricking you into downloading malware or a virus. Make sure before you download from untrusted sites that you have good anti-virus software running to examine the files - otherwise, your free fantasy animated background can turn into a computing nightmare.