How do I upload a .jpg photo of my cat to the board for posting?
You don’t. You upload a copy to a Yahoo! group or photobucket or something like that, and post a link here to that location.
OK. I’ve got them in Photobucket. Do I just copy the URL and paste it in my post?
Yes, just like you’d post a link for anything else. You have posted links before, right? 
you post a link by typing <url=“Uploading photos to the SDMB - About This Message Board - Straight Dope Message Board”>your link here</url>, except you use left and right square brackets rather than <>, so it looks like this.