A view of the diversity of early fish and their relatives. I drew this for a magazine article.
First, the two early animals that are not actually fish, but can maybe tell us something about the origins of fish:
Pikaia is a Cambrian animal that might be a chordate. It looks quite a bit like the modern lancelet, but has weird tentacles around its' head. Pikaia was only about 5 cm long.
Haikouichthys also lived during the Cambrian. It's a craniate, which means it has a skull, but no proper backbone. It's only a few centimeters long and looks superficially like a fish fry, though it doesn't have any paired fins.
Conodonts are a controversial group. They are mostly known for their tooth-like structures known as conodont elements. Rare soft-tissue finds show that they're from the mouth of an eel-shaped animal that seems to have had very large eyes. They were jawless, and the conodont elements formed a very specialized feeding system, though nobody knows what they actually ate with it. They ranged in size from about a centimeter to almost half a meter. The relationship of conodonts and other vertebrates are not clear, though they now seem to be at least proper fish. Conodonts appear in the fossil record around the late Cambrian and lived until the late Triassic.
And then for the less controversial radiation of fish:
Sacabambaspis was a genus of jawless armoured fish from the Ordovician. It had tiny eyes and mouth in the tip of it's snout, and also seems to have had a lateral line system like modern fish. The caudal fin was its' only fin, so it probably wasn't a very good swimmer. It might have swimmed much like modern tadpoles. Sacabambaspis is a part of the Arandaspida, a group of the most basal actual fish.
Pteraspis is part of the group calles Heterostraci, which is closely related to Arandaspida. Pteraspis is also a heavily armoured, jawless fish, but lived in the Devonian. It seems to have been an active swimmer with a streamlined body. Though it had no paired fins either, it had bony "wings" growing from its' armour, which would have acted much like fins.
Asiaspis is a particularly weird genus of Galeaspida, which thrived during the Silurian and Devonian. It's more closely related to the modern fish, but still lacks jaws and paired fins. The function of the spiked head shield is not known.
Cephalaspis is even closer to modern fish. It belongs to the Osteostraci, the sister group of Gnathostomata (which, in turn, includes all the bony fish, cartilaginous fish and tetrapods). It lived during the Devonian and, with it's heavy shields and flattened body, seems to have been a bottom feeder.
Bothriolepis is our first example of a jawed vertebrate. It's a Devonian placoderm, a fish closely related to the modern cartilaginous fishes, but instead of cartilage, it has massive bony armour. Bothriolepis was a hugely successful genus: over a hundred species are known from every continent. It's part of a smaller group called Antiarchi, whose specialty were the pectoral fins that resembled arthropod limbs.
Cladoselache, in turn, is a proper cartilaginous fish related to modern sharks. It's well known because of well-preserved fossils from the late Devonian of North America. Cladoselache had blade-like spined in front of both of its' dorsal fins and it's pectoral fins were huge. Curiously, it was almost entirely naked, since it had to scales or bony armour. Unlike most fishes of its' time, it probably counted on speed instead of armour against predators.