Cladistics is based on the simple facts that the process of diversification follows and “transmission with modification” scheme: a “progenitor” reproduces itself (sexually or not) by transmitting its characteristics to his descendants, by there are some modifications otherwise it would be a mere cloning that would not lead to diversity.
Cladistics proceeds by using the properties of a species that can be traced throughout this process. These properties are called characters.
The characters are divided into states. Basically, there are two kinds of states: the ancestral one, which is characteristic of a common general ancestor, and a derived one, that is a proper innovation. A filiation is then identified with species that share the same innovation. To be somewhat clearer, a species is individualized by an innovation, making it a novelty in the diversification process. Its descendants will inherit this “innovation” which is defined as a derived state for them. All species having this derived state have necessarily the same common ancestor.
Indeed, the species can be several things: “true” species, individuals, or some hypothetical objects. All these are preferably called taxa (taxon in singular).
The resulting classification is hierarchical and is represented on a tree that is called a cladogram. From this tree, one can define clades, which are evolutionary groups of taxa having a same common ancestor. The taxa alway occupy the leaves of the tree, while the nodes are hypothetical and non-indentifiable ancestors. The nodes can be seen as events in which some innovations occured.
The tree can be rooted with a comparison group or outgroup or extragroup. This outgroup shares a common ancestor with the ingroup (the group in study). Rooting a tree orientates the diversification, but finding the correct outgroup is not always easy.
Clades are also called monophyletic groups because they are made of a ancestor and all its descendants. Otherwise we get a paraphyletic group, while if several ancestors are present, we call it a polyphyletic group.
Characters must have states, but these are not necessarily binary nor discrete. Clades are constructed from synapomorphies which are the character states shared by an ancestor and all its descendants. However, characters can behave in a more complicated way and show homoplasies that come from a parallel evolution (innovation that appears independantly in two lineages), a convergence (same character state that results from different evolutionary paths) or a reversal (a character that evolves back to an ancestral state).
The homoplasies, if too numerous, can kill a cladistic analysis, whereas there are heavily used in classifications that rely on similarity as a criterion for clustering.