First go to photobucket.com and create a free account if you don't already have one. Follow the instructions to upload your photo to their server.
It will look like this:

Make sure the button for My Computer is clicked and then click on the Choose File button. This will cause a pop up to access your images on your hard drive. Choose the image and click okay. This will begin the upload process.
Once the image is uploaded the web page will show the new image and you can add a title or what not. Click Save and Continue and the page loads to show your album with the picture you just added.
Once the picture is uploaded, you will see the code used to put it here under a small version of the photo. It will look like this:

Now just click inside the box next to HTML code and it automatically copies the code needed.

To post here just right click and choose Paste to paste the code. Or you can just Ctrl+V to paste. Either way, the code is now pasted to display your image.
Now, all you really need in that code is the img src part, not the a href, the border=0 or the alt text of photobucket. If you know anything about html, you will know what to remove to keep the code from linking back to photobucket. I can show you what to delete here:

That's about it! Good luck!