The "that cat" meme started on the "LMAO Pics V. 76: Idiot Man-Babies Arguing About Thread Titles" thread on popular forum Facepunch. It started a little earlier than that, but "LMAO Pics V. 76" is where the meme started.
It started becoming a meme when facepunch user kubakabana was banned for posting the cat more than once. The ban reason: (User was banned for this post ("Stop posting that cat." – Pascall))
the name "that cat" caught on on the post right below it made by the user hgncommand. After quoting the picture, he wrote "that cat" and it was the name of that picture from that day forward.
On the next page, user funion posted the "that cat" picture and was first banned by mod Asaratha, and then perma-banned by mod Gurant. this started the ball rolling for scattered posts of that cat by brave users. The punishment for posting it was being perma-banned.
"that cat" was photoshoped into many other different memes and is still being posted to this day.