Archosaurs, or "ruling lizards," are an infraclass of reptile, one of three infraclasses of diapsid, or "two-arched," reptiles. Diapsids are one of two classes of reptile, the other being anapsids, meaning "without arches." The difference between diapsids and anapsids is that the latter have two holes on each side of their skulls, near the temples — made to decrease the weight of the skull — while anapsids only have holes for the eyes. Anapsids are represented today by turtles, tortoises, and terrapins, while diapsids make up all other reptiles, as well as birds. It is unknown whether living anapsids descended from anapsid or diapsid ancestors.
Archosaurs, one type of diapsid, are the group most famously known for having dinosaurs as its members. Every dinosaur was an archosaur, from the leaf-eating Stegosaurus to the titanic Brachiosaurus and the deadly Tyrannosaurus rex. Pterosaurs, a type of flying reptile that ruled the skies before birds, were also archosaurs, as are living crocodilians (alligators, crocodiles, and gharials) and all birds (descended from small dinosaurs that survived the K-T extinction event). Archosaurs first evolved in the late Permian (~255 million years ago) or the early Triassic (~250 million years ago), depending on whether you consider the earliest archosaur-like animals to be "archosauriformes" or true archosaurs.
Archosaurs are characterized by teeth set in sockets which makes them less likely to be torn loose during feeding and antorbital fenestrae, or holes in their head, in front of the eyes, to reduce skull weight. They also have mandibular fenestrae, which are small holes in the jawbone, also to decrease weight, and fourth trochanter, a ridge for muscle attachments on the femur. Some scientists consider the last trait to be one of the most important, and casually related to the rise of the dinosaurs and the ability of early archosaurs to survive the most catastrophic mass extinction of all time, the Permian-Triassic extinction, 251 million years ago.
Archosaurs and early mammal-like reptiles (therapsids) evolved at around the same time — the Permian era. Many believe that it could have gone either way, with mammals or reptiles becoming the dominant life form on Earth, but for several reasons, some of which may have been mere chance, the archosaurs came out on top, ushering in the 155 million year-long Age of the Dinosaurs. One reason may have been the previously-mentioned femur ridge, which helped archosaurs to stand more erect, bypassing Carrier's constraint, a rule that states that an animal with sprawling limbs cannot walk and breathe at the same time. However, some paleontologists are skeptical of this explanation because archosaurs were already on the rise when they had sprawling limbs.
In the last 65 million years or so, mammals have become the dominant form of life on Earth, displacing archosaurs. However, they are still all around us, in the form of comparatively unthreatening birds.