Here we've listed the herrerasaurs and other basal saurischians which are (or were once believed) too primitive to be either true theropods or sauropodomorphs. The most commonly-seen in media have been Herrerasaurus & Staurikosaurus (since the 1980s) and Eoraptor (since the 1990s).
Starry dinosaurs: Herrerasaurus & Staurikosaurus
Thanks to dino-books and documentaries like those of the Walking with Dinosaurs series, several people have become conscious about the existence of Coelophysis, which has become “the forerunner of the dinosaur world”. However, some carnivorous dinosaurs lived even before it; but are so ancient, that could not even be real theropods. In Triassic world, dinosaurs still were not so differentiated each other, and the familiar “Coelophysis” shape was shared by several other animals, obviously with some degree of variation.
Herrerasaurus and Staurikosaurus are the two most classic examples. Together, they form their own dinosaur subgroup, Herrerasaurians. Their shape was typically theropodian, but their skeleton was more archaic and less bird-like; for example, they had five digits in their feet, more similarly to sauropodomorphs than to neotheropods (theropods more derived than herrerasaurs), which have only four. Also their pelvis were unique. This bony-puzzle was responsable of many headaches among paleotaxonomists: Herrerasaurus and Staurikosaurus have been variably classified as true theropods, true sauropodomorphs, neither-theropod-nor-sauropodomorph saurischians, or even non-dinosaur dinosauromorphs!
Herrerasaurus was the first discovered, in the 1960s. Found in what is now Argentina, it’s the biggest of the two (4 m [12 ft] long), was longer than a Coelophysis and much more robust, with a larger, stronger head and much shorter neck. It was arguably a more powerful predator, hunting relatively large animals such as rhynchosaurs, basal synapsids and small-sized dinosaurs, but retreated against the giant prosauropod Riojasaurus or the 7 m-long Postosuchus relative Saurosuchus. Discovered in 1970, Staurikosaurus shared the same body-structure but was only the size of an Ornitholestes (2 m [6 ft] long), and arguably hunted smaller preys, perhaps young rhynchosaurs or the primitive ornithischian Pisanosaurus.
Staurikosaurus is one of the most poetically named dinosaurs, “Southern Cross lizard”: it has been for several decades the only dinosaur found in Brazil, and Brazilian flag shows just this constellation. Together, these two dinosaurs have long disputed the title of “the first/most primitive dinosaur ever appeared on Earth”. Among the numerous hypotheses, some paleontologists went to claim herrerasaurians were the ancestors of all the other dinosaurs: now this hypothesis is totally discarded, since both animals had their specializations on their own.
Dawn terror?: Eoraptor
Discovered in Argentina in 1993, Eoraptor (“dawn robber”) suddenly seemed to solve the rivalry between Staurikosaurus and Herrerasaurus for the “Whoa, the very first dinosaur ever appeared!” title. When was described, it was thought more primitive than both; however newer studies don't always agree with this.
3-4 ft long, the same size of a Compsognathus, Eoraptor shared with herrerasaurians some skeletal features resembling non-dinosaurian archosaurs; it too was thought neither saurischian nor ornithischian, but a more basal animal in the middle between true dinosaurs and other dinosauromorphs such as contemporaneous Lagosuchus (also found in South America).
Eoraptor has been the most celebrated among all the supposed “first dinosaurs”, in part because was discovered just at the time Jurassic Park came to audiences - and the fact scientists gave to it the now-familiar suffix “-raptor” could have done its bit, too. Since then, our “dawn robber” has gained much attention in media, also being object of some degree of sensationalism. Several awesome nicknames were invented, from the first terror to The father of all killer dinosaurs.
But Real Life Eoraptor wasn't so fearsome, really: it was a tiny, gracile dinosaur, which could even become a meal for a hungry Herrerasaurus or even a Staurikosaurus. Adding to this, its unspecialized teeth were more probably from an omnivorous rather than carnivorous animal. Science Has Marched On Even More as recently as in year 2011, and one study has found Eoraptor to be a very unspecialized sauropodomorph. Good-bye, “first-terror”.
Who's really the forerunner?: Nyasasaurus & Guaibasaurus
Giving the coup-de-grace to Eoraptor, its common-ancestor-of-all-dinosaurs title is now contended by other dinosaurs found and/or described in the 2000s. Astonishingly, many of them come from South America like the examples above, to the point South America could be renamed “the cradle of the dinosaur kind” at this point — an exception is Nyasasaurus (arguably the very first dinosaur), found in Africa in 1956 but described only in 2012. Other exceptions are the North American Chindesaurus (possibly a herrerasaur) and the Indian Alwalkeria (possibly non-dinosaurian and/or a chimaera). The most basal dinosaur was the common ancestor of Saurischians and Ornithischians, and almost certainly was more similar in anatomy to the former than to the latter: among archosaur groups saurischians are less-derived than ornithischians, at least if you count the structure of the jaws and the pelvis.
Some basal saurischians make together the Guaibasaurids. These saurischians were extremely generic and unspecialized dinosaurs, whose external shape was really in the middle between a small theropod and a "prosauropod", not deceptively theropodian like herrerasaurians or Eoraptor. The namesake Guaibasaurus was the first discovered and initially considered a possible basal theropod (which it may actually be); then, the mythical-named Saturnalia note from Brazil, which was initially believed the “first prosauropod'”.
In the second half of the 2000s, scientists decided that both dinosaurs were too basal to be either theropods or sauropods, and put together in the same family, guaibasaurids. Now they are often considered very basal sauropodomorphs, though this, too, may still change (for instance, Guaibasaurus itself may be a theropod). With their unspecialized traits, guaibasaurids were almost surely omnivorous creatures; indeed, a third member found in 2009 has received a meaningful name: Panphagia, “eat-all”.