Taxonomy is the study of how living organisms are related to each other, and what makes each type different from all the others. It's basically trying to draw up a family tree of all living organisms. If we are dealing with modern animals we have many ways of looking at these relationships, the most important of these being genetics. Plesiosaurs are extinct, and we have no way of looking at their genes, so we have to base our family tree on skeletal morphology - the shapes of the bones.
There's a simple story, and many more complicated versions.
The simple story goes like this: There are two kinds of plesiosaur, one small-headed, long-necked, and the other large-headed and short-necked. The large-headed type is called the pliosaurs. There are three kinds of long-necked types, the plesiosaurids, the cryptoclidids and the elasmosaurs. Of these, the plesiosaurids are the earliest, and it is thought that all the other forms including the pliosaurs are descended from them. Their remains are found in rocks from the Upper Triassic and Lower Jurassic, between 220 and 175 million years ago. They had fairly long necks - up to 32 vertebrae. The elasmosaurs and cryptoclidids appeared in the Upper Jurassic, about 160 million years ago, and survived until the end of the Cretaceous, 65 million years ago. Elasmosaurs had very long necks - up to 72 neck vertebrae. Cryptoclidids had relatively shorter (though still long) necks, and smaller, more slender teeth. They lived at the same time in the same places, but fed on different prey.
Pliosaurs first appeared in the Lower Jurassic, 200 million years ago, and died out in the Middle Cretaceous, about 80 million years ago. They were ferocious predators, and the biggest were enormous - up to 17 meters long, and weighing probably over 10 tons. They had huge, conical teeth and fed on anything they could get their teeth into, including the other plesiosaurs.
The more complicated versions are - more complicated. They start like this: Plesiosaur taxonomy is a mess.
Plesiosaurs were some of the earliest extinct animals to be discovered, and in the early days their morphology was not well known. New species were erected on the basis of fragmentary material, and initially most were assigned to the genus Plesiosaurus. We are still trying to untangle the confusion this caused. It is clear that in some cases different parts of the same species were described under different taxa, and that many taxa were erected on the basis of non-diagnostic material. Arguably the worst proponent of this was Richard Owen, who erected on fragmentary remains almost 100 species of the genus Plesiosaurus. Very few of these still stand.
Plesiosaurs lived in water. The body shape of aquatic animals tends to be strongly convergent, and it is not easy to distinguish features which are a requirement for a particular form of locomotion or behaviour in a marine environment from those which show evolutionary relationships. Although plesiosaurs are known from a large number of specimens, a lot of these come from a few deposits. A substantial body of knowledge has been built on the material from the Oxford Clay of South-east England, largely through the Leeds collection - perhaps the greatest collection of vertebrate fossils ever built up by an individual or family. It is not known how typical these specimens are of plesiosaurs in general - they may be the equivalent of kangaroos!
The model we have of plesiosaur evolutionary relationships is built by looking through a few small windows, widely separated by time, not knowing if what we are looking at is typical of the family as a whole, or a marginal group. The fundamental question is this: Are the two basic plesiosaur morphologies taxonomic groups, or are they morphotypes? In other words, are the long-necked forms more closely related to other long-necked forms, and the large-headed forms more closely related to other large-headed forms, or do evolutionary pressures push a plesiosaur lineage towards one or other of the two forms?
The picture emerging from several strands of research is that plesiosaur evolution is much more complicated than we thought. It seems possible that the elasmosaurs for example are not a single lineage but as many as three not necessarily closely related lineages. Some pliosaurs families appear to be more closely related to the long-necked forms than to other pliosaurs.