So I notice alot of people with animated/gif style avatars, and I was wondering how you guys make that happen.

Just put a gif as the avatar. Just copy the link to it and paste it to the avatar text box.

Can use software that supports multiple frames and other animations. Or a lot of people just place images into a gif maker website.

Yes but how do I do this without hotlinking?

You have to put them on sites such as imgbb(the one that I use) and others like it ^.^

Ok I know that you can post images on those sites, but I'm having trouble posting a gif :/

I had to use a different one but I can't think of it. At the moment.

Try uploading on Postimage