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The best-known kind of taxonomy is used for the classification of lifeforms (living and extinct). Each organism has a scientific name. This name is part of the biological classification of that species. The name is the same all over the world, so scientists from different places can understand each other. In addition, a species has a position in the tree of life. Thus the crow is Corvus corone, a member of the Corvidae family, and they are passerine birds. That is well agreed, but the classification of some groups is not agreed at present, and often several classifications are being discussed.
Living things are classified into three domains: bacteria, archaea and eukaryotes. The highest rank in a domain is the kingdom. Each kingdom has many smaller groups in it, called phyla. Each phylum has more smaller groups in it, called classes. This pattern looks like branches on a tree with smaller branches growing from them. Each species is put into a group because of what it does, how and what it eats, special body parts, and so on. At the end of the pattern, the groups (genera) are very small. Then each species in the genus is given its own name.
When someone writes about a living thing and its formal scientific name, they write the genus and species name. This is known as binomial nomenclature, because it uses two names for each organism. The first is the genus name, and the second is the species in that genus. The scientific name of the domestic cat is Felis catus. Sometimes it is enough to write F. catus.
When people started naming species, Latin was a language widely used in Europe. All species names are still written in Latin. This has some advantages. Since Latin is no longer spoken, it is unchanging, and is owned by no-one. It gets over the problem of every language having its own names for animals and plants.
Scientists used to write the official description of each new species in Latin. On 1 January 2012, the International Botanical Congress changed to allow English (as well as Latin) for describing new plant species. The International Code of Zoological Nomenclature recommends choosing a language that is widely used, and that is used in the places where the species lives.
An important modern approach to taxonomy is cladism. This approach is based on the branching (tree-like) course of evolution. Like traditional Linnaean classification, it uses traits to decide on the branches of the classification. It insists on groups being monophyletic. This has the effect that birds are not a class but a sub-group of dinosaurs. It also means the ranking system described above would be abolished.
So cladism has different principles of taxonomy, and produces a different kind of taxonomy. Decisions, where possible, are supported by DNA sequence analysis. Present-day biological classification is a mixture of the old Linnaean and the modern cladistic principles of taxonomy. In parts, it is changing rapidly. The classifications presented in Wikipedia at present are often a compromise between the two systems. The details are regularly discussed.
Turmoil in taxonomy
Today, there are many changes in the classification of living things. This turmoil in taxonomy has led to many alternative classifications. It is caused partly from the move from Linnaean to cladistic principles, and partly by the use of DNA sequence data in taxonomy. An example is: the way derived groups like birds should not be classified at the same level as the group they evolved from. Yet birds have traditionally been a class under the Linnaean system.
The turmoil sometimes results in differences between related pages. Pages may rely on different references and different authors' opinions as to the present best arrangement.
The following source is good on the differences between cladistic and taxonomic classification systems:
- Grant, Verne 2003. Incongruence between cladistic and taxonomic systems. American Journal of Botany. 90 (9) 1263-1270. 
- Molecular clock
- Molecular evolution
- Sequence analysis
- Biological classification
- Military taxonomy
- Taxonomy Citizendium