This sail-backed predator is by far the most iconic of the synapsids. The obvious mascot for the Permian Period (the football team in Friday Night Lights really ought to be named for this bad boy). I fondly remember the blue-green Dimetrodon model I had as a kid, which came with a little club-wielding caveman. I assume he was included for scale, though if I remember correctly, Mr. Club was way-hay-hay too small - he may have reached the dimetrodon's chin. He should have been about half as long as the beast. Inaccurate model kit aside, dimetrodon is one of the most common pseudo-dinosaurs; in fact, google auto-fill just listed "dimetrodon dinosaur" as its second option. I'm assuming those options are ranked by frequency of searches.
As you can tell from the posture of the legs in the above image, dimetrodon has a classic sprawling leg like the reptiles. Only dimetrodon isn't a reptile, either. Dimetrodon was a synapsid, which used to be called the "mammal-like reptiles" for a reason: they are an early part of the line that eventually spawned the mammals. Dimetrodon's family are the most derived (a less loaded way of saying "advanced") synapsids and thus are the most closely related to the mammals. After thousands of generations, their siblings the therapsids would eventually bring about the mammals, who trolled around in the shadows as the dinosaurs had their way for a hundred and fifty million years or so. But by that time, dimetrodon was long gone. (Dimetrodon illustration by D. Bogdanov, via wikipedia)
Plesiosaurs, Icthyosaurs, and Mosasaurs
These, the most famous of the ancient marine reptiles, are also very frequently called dinosaurs. Plesiosaurs were long-necked, short tailed reptiles; popular depictions of the Loch Ness monster usually resemble them. Icthyosaurs are more "dolphin-like" in appearance, with slender snouts and long tails. Mosasaurs are just frickin' huge sea monsters, really, and a member of the proud tradition thatwould produce modern snakes and lizards: they are called squamates, and their scales overlap.They are all aquatic diapsids, reptiles who became adapted to a marine lifestyle but are descended from landlubbing ancestors. It's pretty easy: if it has flippers, it ain't a dinosaur. Dinosaurs didn't live in the water. (Illustrations of Plesiosaurs and Icthyosaurs by Heinrich Harder, public domain. Mosasaur by Arthur Weasley)
Mammoths and Glyptodonts
The mammoths look familiar enough, being big, hairy elephants, that it's only rarely that I see them called them a dinosaur. Glyptodonts, though... I've seen it happen quite a few times. I suppose because of their body plan, which superficially resembles the armored, club-tailed ankylosaurs. Well, they're mammals, so they're more closely related to dimetrodon than any dinosaur, and they lived a great many millions of years after the Cretaceous era, and the dinosaurs' reign, ended. In fact, the've only been gone for about 10,000 years, and our own ancestors walked among them in South America, using those big domes to create the Miocene version of suburban cul-de-sacs (JK, though they did use them as shelter). They actually are distant relatives of today's armadillos, so those of us who can't wrap our heads around the fact that superficial resemblance doesn't always correlate to evolutionary affinity can smile at that small victory. The Germans, by the way, call armadillos panzerschwein, which means armored pig, and that is totally boss. If you ever want to spend some good quality time with some panzerschwein, head down to Cumberland Island National Seashore. The place is lousy with 'em. (Illustration of Glyptodon and modern humans by Heinrich Harder, public domain)
Any damn thing with a name ending in "-saur"
It just means "lizard," people.