A summary of this scheme, from most general to most specific:
A strength of Linnaean Taxonomy is that it can be used to develop a simple and practical system for organizing the different kinds of living organisms. The most important aspect of this is the general use of binomial nomenclature, the combination of a genus name and a specific epithet (tuberosa, in this example), to uniquely identify species of organisms. In the example above, the butterfly weeds are uniquely identified by the binomial Asclepias tuberosa. No other species of plant can have this binomial. In this way, every species can be given a unique, stable name.
Rules for the proper naming and classification for all types of living organisms under the Linnaean system have been adopted by professional biologists. The rules governing the nomenclature and classification of plants and fungi are contained in the 'International Code of Botanical Nomenclature,' maintained by the International Association for Plant Taxonomy. Similar codes exist for animals and bacteria. Scientists follow these codes so that the names of organisms can be as clear and stable as possible.
Over time, our understanding of the relationships of living things has changed. The greatest change was the widespread acceptance of evolution as the mechanism of biological diversity and species formation. After this, it became generally understood that classifications ought to reflect the phylogeny of organisms, where each taxon should originate from a single ancestral form. In some systems it is generally encouraged that taxa should be strictly monophyletic, but this is controversial.
Originally Linnaeus had three kingdoms in his scheme, namely Plantae, Animalia and an additional group for minerals, which has since been abandoned. Since then various forms have been moved into three new kingdoms - Monera, for prokaryotes, Protista, for protozoans and algae, and Fungi. This scheme is still far from the phylogenetic ideal and the five kingdom view has largely been supplanted in modern taxonomic work by a division into three domains - Bacteria and Archaea, which contain the prokaryotes, and Eukaryota, comprising the remaining forms. This was precipitated by the discovery of the Archaea.
See also evolutionary tree, which has some further subdivisions and presents the current taxonomic view.