KnowYourMeme says that the term "LOLcats" wasn't coined until 2007* when someone pasted a caption saying "I CAN HAS CHEEZBURGER?" (with the classic Impact font) onto a photo of a British shorthair named Frank, on the 4chan chat website.
TheStar explains:
Some time in 2005,* an anonymous poster put text over a picture of a very relaxed cat, indicating he was waiting for "Caturday." Soon, 4chan.org was flooded with pictures of cats in various states of idyll proclaiming their love for "Caturday."
Because it was more cute than funny, it would have been a short-lived fad had not another anonymous poster uploaded a picture of a hungry-looking, plus-sized grey cat with the text, "I can has cheezburger?" It was followed by a picture of the same cat snarling at a heap of clementines, with the words "Do not want!"
And thus LOLcats and memes were born.
*It's not clear if 2005 was the correct date, or how much later the "Cheezburger" photo was posted, but a graph by Google Trends of some early LOLcat search terms shows that the term "LOLcat" itself wasn't used until 2007.
The photo itself was from much earlier -- Frank's disembodied cat head had also been used in 2003, on the website of a Russian pet food company, advertising "Happy Cat."
It's not clear who took the original photo, but KnowYourMeme.com says that Frank was born in 1994 and died in 2007.