There are several ways. You can get a free account at Photobucket, Imageshack, or any number of other image hosts. Google "image host" and you'll get a bunch of choices.
Once you have your account, you can upload whatever photo/gif you want to your account. Both Imageshack and Photobucket generate code for forums, so you just copy/paste the code they give for your image into your post, and you're done.
You can also use the Attach Files button below the text box, and upload your image directly to HuskerBoard. If you're using the Quick Reply box below the forum, hit the More Reply Options on the bottom right to see the Attach Files tool. Hit the Choose Files button and it'll walk you through the process. Note that you have just under 60MB of total space, so if you use that option regularly you'll run out of space after a short time.
Your avatar seems to be broken right now. To change that, go to your profile and click on your avatar. A hover button named "Change" will appear. Click that button and walk through the steps there to change your avatar.