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A taxon (plural taxa) is a taxonomic group or taxonomic unit. It is a group of organisms which a taxonomist decides belong together.
A taxonomist can assign a scientific name and rank to a taxon; this places it at a particular level in a hierarchy. It is not strictly necessary to assign either a name or a rank to a taxon, but doing so makes it much easier to refer to the taxon. However, many taxa have to wait for years before getting a name.
There is no limit to the number of ranks that may exist, but the most important ranks are, in hierarchical order:
A simple phrase to remember the order is "Dignified Kings Play Chess On Fine Green Silk" (there are many other such phrases).
A distinction is made between taxonomy and systematics. Systematics deals with how groups relate to each other. Taxonomy deals with making groups (taxa), deciding what belongs together. Taxonomy is also known as classification.
A special part of taxonomy is nomenclature. This consists of rules on what names to use. So a taxonomist first decides what does and does not belong in a group, and then uses nomenclature to decide what name this group should have. If a group is made larger or bigger it may get a different name. On the other hand, the same name may refer to a bigger taxon (according to one particular taxonomist) or a smaller taxon (according to some other taxonomist). This means that scientific names are not guaranteed to be stable. Most names are stable, but for some taxa there is no agreement on its name, because taxonomists do not agree what does belong together and what does not belong together.
A taxonomist can decide for himself what scientific reasons he adopts for making a group (a taxon). If he is not convincing in his choice, other taxonomists will not agree with him, and they will then make other arrangements. These days biological classification is mostly supposed be done according to evolutionary (phylogenetic) relationships, so far as these are known.