It's very simple. Create an account at a free image host. Some free sites are:
Photobucket
Image Shack
Image Bam
Tiny Pic Note: You don't need to create an account, but you will have to remember the URL of your image.
There are dozens of others out there, each one will probably have a limitation on their free service, so be sure to pick the one that's right for you.
Now....
The image host will give you an option to upload your picture. (See below.)

If they're like Photobucket, you'll also see the IMG options displayed below the picture you've uploaded..

Now, in the forum post options, use the IMG and put in the URL -web address- of your image. (Or just copy and paste the IMG tag from your image server.)

(If you use Photobucket, you can also automatically log in by clicking the Add Image or Video from Photbucket.com link. You'll have to enter your username and password for Photobucket.)
The code should look like this:
[img]http://i194.photobucket.com/albums/z132/santas_lyssa/ForumPosts/forumupload.jpg[/img]
Viola! Your pic should display in your post. Photobucket does get quirky sometimes, so hit the "Preview" button (next to 'submit') before you post.
Let me know if you need any help.