Well, I will answer your question and then give you my UNofficial answer for Eurobricks on the subject.
Moving images are referred to as .gif files instead of .jpg files. If you have Photoshop, that comes with a program called imageready which allows you to make .gif files. Otherwise you can search them out on the internet, like so.
[img=[b]http://th181.photobucket.com/albums/x185/rockstarosa/myspace/th_lego_dance02.gif[/b]]
You then would proceed to upload that file as your avatar.
NOW, there is a limit to 100k for avatars which is kind of small when it comes to animated .gif files, ALSO I am fairly certain and have heard in the past that EuroBricks Staff would prefer avatars to NOT be animated.
It does not specifically say avatars here, but they do adress it in the signatures section, so I would take that as a no for avatars as well, plus I know I've seen an admin address it before.
Signatures: As a rough rule, you may use up to four pictures each with a maximum pixel height of 200, a file size of 250k and no more than 200 words. If your sig is bigger than your average post, it can confuse someone reading a thread and you may be asked to reduce it, or a staff member may do it for you. Some exceptions may be made on a case by case basis, but please try to limit the space you use. Font size should be kept to a minimum: ideally no bigger than size 3. No animated GIFs are to be used in your signatures. <<<<Link to Guidelines thread