A taxon (plural: taxa) is a group of (one or more) organisms, which a taxonomist adjudges to be a unit. Usually a taxon is given a name and a rank, although neither is a requirement. Defining what belongs or does not belong to such a taxonomic group is done by a taxonomist. It is not uncommon for one taxonomist to disagree with another on what exactly belongs to a taxon, or on what exact criteria should be used for inclusion.
Taxonomists sometimes make a distinction between "good" (or natural) taxa and others that are "not good" (or artificial). Today it is common to define a good taxon as one that reflects presumptive evolutionary (phylogenetic) relationships. But this is not mandatory.
A taxon may be given a formal name, a scientific name. Such a scientific name is governed by one of the Nomenclature Codes, which sets out rules to determine which scientific name is correct for that particular grouping.
Advocates of phylogenetic nomenclature, using cladistic methods, do require taxa to be monophyletic, consisting of all descendants of some ancestor. They generally do not refer to taxa as their basic unit, but to "clades", a clade being a special form of taxon. However, even in traditional nomenclature, few taxonomists of our time would establish new taxa that they know to be paraphyletic. A famous example of a widely accepted taxon that is not also a clade is the "Reptilia".
But there are other definitions.
A taxon can be assigned a rank, usually (but not necessarily) when it is given a formal name. The rank of a given taxon is not necessarily fixed, but can be altered later by another (or the same) taxonomist.
A prefix is used to indicate a ranking of lesser importance. The prefix super- indicates a rank above, the prefix sub- indicates a rank below. In zoology the prefix infra- indicates a rank below sub-. For instance:
Rank is relative, and restricted to a particular systematic schema. For example, liverworts have been grouped, in various systems of classification, as a family, order, class, or division (phylum). The use of a narrow set of ranks is challenged by users of cladistics; for example, the mere 10 ranks traditionally used between animal families (governed by the ICZN) and animal phyla (usually the highest relevant rank in taxonomic work) often cannot adequately represent the evolutionary history as more about a lineage's phylogeny becomes known. In addition, the class rank is quite often not an evolutionary but a phenetical and paraphyletic group and as opposed to those ranks governed by the ICZN, can usually not be made monophyletic by exchanging the taxa contained therein. This has given rise to phylogenetic taxonomy and the ongoing development of the PhyloCode, which is to govern the application of names to clades.
A taxon (plural taxa), or taxonomic unit, is a name that means an organism or group of organisms.
One has to distinct between taxa/taxonomy and classification/systematics. Taxonomy refers to biological names and the rules of naming. Classification refers to rank ordering of taxa according to presumptive evolutionary (phylogenetic) relationships.
A broad scheme of ranks in hierarchical order:
A simple phrase to remember the order is "Dignified Kings Play Chess On Fine Green Silk".