Therapsids (meaning "beast face") is a class of synapsids, animals dominant on Earth during the mid to late Permian period (about 300 to 251 million years ago). Synapsids themselves are one of two major groups of amniotes (egg-laying animals), the other being sauropsids, or the true reptiles. Non-therapsid synapsids, the pelycosaurs, were actually more dominant during the Permian, but the therapsids are significant because they eventually evolved into mammals, survived the Cretaceous-Tertiary extinction that killed the dinosaurs, and went on to become the dominant group of terrestrial vertebrates, culminating in the evolution of Homo sapiens.
Animals in this class emerged during the early Permian, and are considered part of the first wave of amniote diversity. Some of them, like the cynodonts ("dog teeth"), remarkably resemble modern-day animals, including differentiated teeth, a bulging braincase, and walking in an upright manner (not bipedal but upright, distinctive from reptiles). The primary difference is that all therapsids laid eggs. Historically, they were called "mammal-like reptiles," but now they are recognized as distinct from reptiles, and are considered basal mammals instead. These animals are a very confusing area of vertebrate evolution, and much less popularized than the dinosaurs or crown group mammals, but necessary to understand the full picture of the evolution of life on Earth.
There were at least three major clades of therapsids: the dinocephalians ("dinosaur heads"), the least advanced therapsids; the herbivorous anomodonts (beaked herbivores), which were the most diverse and numerous of the Permian herbivores; and the mostly carnivorous theriodonts, the most mammal-like of the bunch, and the ancestors of all modern mammals. Another group, the carnivorous biarmosuchians, are either a paraphyletic clade — consisting of several unrelated groups — or consist of a basal offshoot of other therapsids. As you can tell, the situation is very confusing.
During the end-Permian extinction 251 million years ago, most of the therapsids were wiped out, along with 95% of all other animal species. For most of the next 190 million or so years, throughout the Mesozoic period, they would be lost in the shadows of the dinosaurs — the dominant form of life on Earth. Only after all the non-avian dinosaurs were killed out did they reemerge as mammals.