Synapsids include mammals and our distant ancestors, including pelycosaurs and therapsids, while sauropsid is another word for reptiles. Synapsid means "fused arch," a reference to skull structure. Another name for a synapsid is theropsid, which means "beast face," in contrast to sauropsid, which means "lizard face." Synapsids are sauropsids are the two evolutionary lineages of amniotes, which includes all non-amphibians tetrapods and their descendants (such as whales, which descended from tetrapods but lost their legs when they became exclusively marine). Early synapsids used to be called "mammal-like reptiles," but this is a misnomer, as they were not reptiles at all.
Synapsids and sauropsids split off from each other approximately 320 million years ago, during the late Carboniferous period. Both looked like small lizards. At the time, tetrapods had existed in the water for about 45 million years, and on land for at least 20 million years. Both are amniotes, that is animals with complex eggs that can be laid on land, in contrast to amphibians, which must lay their eggs in water. Before synapsids and sauropsids split, there were some stem-group amniotes that didn't fit into either group. Amniotes were destined to inherit the Earth because they are the only land vertebrates that can venture significant distances from water and still survive.
The difference between sauropsids and synapsids is defined in terms of the openings in their skull. Synapsids have an extra hole, used to reduce skull weight and provide an attachment point for jaw muscles. Sauropsids began with no holes in their skull, then developed one pair, with each hole behind the eyes. Initially, both groups were "cold-blooded" (ectothermic).
Since the late Carboniferous, the colonization of the land by large creatures has been an evolutionary arms race between synapsids and sauropsids. Synapsids got off to a great start, diversifying more rapidly than sauropsids and giving rise to most of the large animals of the Permian, including the successful pelycosaurs, some of which were as big as trucks, and had the only apex predators of the time.
At the end of the Permian, the largest synapsids went extinct, leaving many niches open for exploitation. The sauropsids took advantage, eventually giving rise to dinosaurs, which dominated the Earth throughout the Mesozoic. About 65 million years ago, the tables turned again, when an asteroid wiped out all non-avian dinosaurs. Synapsids ruled the world again, in the form of mammals. Eventually, synapsids gave rise to humans, arguably the most evolutionarily successful terrestrial vertebrate in the history of life on Earth.