Deuterostomes are a large superphylum of animals that includes chordates (vertebrates), echinoderms, hemichordates (acorn worms), and one small phyla, Xenoturbellida, which consists of two marine worm-like species. "Deuterostome" means "second mouth organism," a reference to the fact that during embryological development in deuterostomes, the first body opening becomes the anus, and the second the mouth, unlike in protostomes, where it is the opposite. Another difference is in the cleavage of the embryo during growth -- deuterostomes display indeterminate cleavage, where their is more room for variation in where cells ultimately end up, in contrast to the protostomes' determinate cleavage, where the end position of every cell is fixed.
The smaller superphylum relative to protostomes, deuterostomes have the benefit of containing the most complex animals. All animals above a certain arbitrarily-defined level of complexity are likely to be deuterostomes. For instance, arguably the most complex phyla, the Chordates, are all deuterostomes. Unlike protostomes, which often have a very small body cavity (coelom) or none at all, deuterostomes feature more prominent coeloms, which helps provide a framework for muscles to pull against, as well as permitting the development of more complex organs whose structural evolution is independent from that of the body wall.
Many of the simplest deuterostomes have three simple features -- gill slits, a hollow nerve cord, and a segmented body. It seems likely that the common ancestor of all deuterostomes had these features. There is some uncertainty as to when deuterostomes branched off from protostomes, but many paleontologists consider 558 million years ago to be a likely lower bound. Ernettia, a fossil from the late Ediacaran period, seems to be a likely deuterostome, while the mysterious extinct phylum from the Early Cambrian, Vetulicolia, have features that suggest they were probably deuterostomes.
Within just 20 million years or so, the earliest simple deuterostomes evolved into the first vertebrates -- especially jawless fish, which are the ancestors of all fish, sharks, and terrestrial vertebrates. Many of the spinoff groups still exist today -- Xenoturbellida, acorn worms, tunicates, lancelets, etc. Tunicates are one of the most unusual of these groups as their larvae shows since of a nerve cord, but they lose it in adulthood. This brings up the possibility that vertebrates evolved from the larval form of certain invertebrate animals.