It's a fairly standard panel of Gary Larson's comic The Far Side from 1982: a caveman stands in front of a large picture of the back half of a dinosaur (likely a stegosaurus from the large plate-like things sticking out of its back). He points to the tail, which has a grouping of spikes sticking out near the end, and says to his audience (in the caption), "Now this end is called the thagomizer, after the late Thag Simmons."
However, scientists have often been fans of The Far Side; Gary Larson has had a newly discovered species of louse and another of a butterfly named after him by fans. So it's not all that surprising that the name "thagomizer" caught on among paleontologists to describe the tail spikes of stegosaur-type dinosaurs, since this is not a feature that is found in other animals and it doesn't seem to have had a name before the cartoon. Ken Carpenter, a paleontologist at the Denver Museum of Nature and Science, is credited with first using the term in 1993, according to the 8 July 2006 issue of New Scientist. But judging by Google searches (10,300 for "thagomizer," and 22,500 for stegosaur tail spike -thagomizer) it is not a universally used term.
Various species of the stegosaur family had from four (the most common) up to fourteen spikes in the thagomizer, arranged in pairs, and each spike could be a meter long. The spikes were as thick as a human wrist at the base and tapered to a point, and composed of bone covered with keratin, the same material as horns and claws. Older depictions of the stegosaurs usually have the spikes pointing upward, but current thinking suggests they should point sideways and slightly backward; the stegosaurs seem to have been only able to swing their tails about 13 degrees to the left or right but even less of a range vertically. The spikes were probably enough to punch through the hide, flesh, and down to the bone of an attacking dinosaur (an allosaurus tail bone has been found with a silver dollar-sized hole in it that might have come from one of these spikes). One problem for the stegosaur defending itself with its thagomizer was that the spikes could snap off and the resulting wound become infected, though a spike that broke at the tip could probably heal over easily.
Sources:
http://www.newscientist.com/channel/life/mg19125592.200.html
http://www.nmnh.si.edu/paleo/dino/stegdcvr.htm
http://dml.cmnh.org/2006Mar/msg00405.html
http://www.translatum.gr/forum/index.php?topic=4401.msg26592
http://forum.wordreference.com/archive/index.php/t-208540.html
http://www.salon.com/people/bc/1999/12/21/larson/index1.html
http://www.palaeos.com/Vertebrates/Lists/Glossary/GlossaryT.html
http://keepleft.org/?q=node/401
http://amos.indiana.edu/library/scripts/tail.html
http://www.findarticles.com/p/articles/mi_m1200/is_17_162/ai_94011334
https://scientists.dmns.org/sites/kencarpenter/PDFs%20of%20publications/allo-stego.pdf