…or at least define enough terms for me to understand what the hell people are talking about when they mention cladistics and so forth?
A good primer for cladistic methods and terminology can be found in this document: The Compleat Cladist (it’s a .pdf document, and I think this is only the first chapter).
Some key terms used in cladistic taxonomy are:
Clade: A monophyletic group
Monophyletic: A group which includes an ancestral species, and all its descendent species. An example would be the clade Dinosauria. For Dinosauria to be monophyletic, it must include all members of Ornithischia and Saurischia, and birds.
Polyphyletic: A non-monophyletic group in which the ancestor of a group is placed within a different group. An example would be Pachyderms, or “herps”. These are groups composed of members belonging to multiple clades, but do not nclude the common ancestors of those clades.
Paraphyletic: A non-monophyletic group in which one or more descdendants are excluded from the group. An example would be the Linnaean class Reptilia. Since the consensus is that birds evolved from reptilian ancestors, Reptilia sans Aves is paraphyletic.
The above refer to relationships between groups. The methods used to determine groupings are typically based on the assessment of various traits, or characters, of species. Some definitions relating to these character assessments are:
Plesiomorphic: A character which exists in an ancestral species.
Apomorphic: A character which exists in descendant species.
The essence of cladistics is in determining whether a trait is plesiomorphic or apomorphic. There are more definitions in the linked document, but the above are the ones you are most likely to encounter.