The stem group/crown group terminology was invented to classify the relationship of living and extinct organisms by Willi Hennig, a German taxonomist, the father of cladistics, in the late 1940s. It was part of his "theory of phylogenetic systematics" that revolutionized the way biologists and paleontologists look at life.
The terms are defined as follows. A crown group includes all living species of the group, plus all extinct descendants back to the common ancestor of all living species. The stem group includes all species not part of the crown group. By definition, every member of the stem group must be extinct. If they weren't extinct, they'd be defined as part of the crown group.
Stem group animals, like those represented by the numerous early tetrapod, mammal, and reptile fossils that have been dug up, give us important information about the course of evolution and how animals tried out different strategies to adapt to their environments. Stem groups are necessarily paraphyletic, meaning they may be more complex than containing just a certain species and all its descendants. A stem group may contain several early offshoots of a group, only one of which went on to evolve into the crown group.
Stem groups are quite common in paleontology. One example would be the stem group mammals, or synapsids — though mammals are technically synapsids too, since they descended from them — which were called "mammal-like reptiles" until it was realized they weren't reptiles at all, but stem group mammals. The earliest synapsids were called "naked lizards" because they would have resembled lizards in appearance, but without scales. As synapsids continued to evolve, they acquired a more mammalian appearance. Sometimes the synapsids are called "basal mammals" or "stem group mammals."
Fish have several stem groups, including the acanthodians, or spiny sharks, and placoderms, or armored fish. Acanthodians, despite their name, are considered closely related to the ancestors of bony fish, while placoderms, more closely related to the ancestors of modern sharks, included the first vertebrate superpredator, Dunkleosteus telleri, which measured 8-11 m (26-36 ft). Both these groups lived during the Paleozoic era, about 300-400 million years ago.