I got asked this question on another forum so I put together this explanation.
The scientific name assigned to each living species is absolutely unique. Common names frequently vary so they are not reliable for identification. Every living organism has only one unique and internationally recognized scientific name.
A binomial nomenclature system is used to name all life, botanical and zoological; simply put, “nomenclature” means the names along with the system used to assign those names, and “binomial” means two names. These two are the genus (plural genera) and the species (or specific epithet). This system was developed by a Swedish botanist, zoologist and physician named Carl Linnaeus who lived from 1707-1778. In 1735, Linnaeus published his Systema naturae per regna tria naturae, secundum classes, ordines, genera, species, cum characteribus, differentiis, synonymis, locis; in translation, “System of nature through the three kingdoms of nature, according to classes, orders, genera and species, with [generic] characters, [specific] differences, synonyms, places.” Usually referred to as simply Systema naturae, by the thirteenth edition in 1767 it had become a monumental classification of all then-known species of life on earth. The system further developed into modern Linnaean taxonomy, the hierarchically-organized biological classification that is today used to classify all species of animals and plants. Strict rules govern this system, established and enforced by the International Code of Zoological Nomenclature [ICZN].
The scientific name (genus and species) is the last and most specific in the hierarchy of scientific classification. The genus is part of a Family, and the Family belongs to a certain Order; for our purposes, we do not need to go higher than the Order. The Family and Order can each be further divided into “Super” and “Sub” families and orders. Each of these terms includes “clades” or clusters of fish that are phylogenetically related. Phylogeny is sometimes referred to as the natural relationships and is an attempt to construct the history of all life based on the evidence from both living and fossil organisms. When classifications are based on phylogenies we can ascertain (and predict) how that group of related fish function, and since this tells us something about their behaviours and requirements it is of interest to aquarists.
Defining “species” is very complex; for our purposes, we may simply consider that the species is the individual fish that is unique from all other fish species. When two or more species are phylogenetically related they will be combined in the same genus. As an example, all fish in the catfish genera Aspidoras, Brochis and Corydoras have a very similar general appearance, easily recognized; but beyond this they share phylogenetical relationships more closely than they do with any other species. Certain specific relationships exist within the species, certain other specific relationships exist between all the species in the genus, and still other relationships exist between all three of the genera within the Family. As an example, Brochis splendens and Corydoras aeneus are almost identical in colouring and pattern; but they are in separate genera because of the rays in the dorsal fin; all Corydoras species have seven rays in the dorsal fin, while all Brochis species have more than ten rays in the dorsal.
Anyone may describe a new species and name it, but this is usually the work of trained scientists with experience as ichthyologists; the name must not be one that has already been used for any species in the same genus, and it cannot be the name of the person doing the naming. The first published name assigned to a new species has priority and remains the valid name with respect to the species; any subsequent species name given to the same species is invalid and may be termed a synonym once it is determined to be the same species. The species name is always the first name that was published for that species, and this name can only change under a few very strictly-enforced rules of the ICZN. The genus may change, sometimes many times, as the result of new scientific study. The Serpae Tetra for instance, Hyphessobrycon eques, has been assigned to five different genera since it was first described in 1882 by Steindachner as Cheirodon eques, but the species name “eques” never changes.
The genus name is either Greek or Latin, and is always capitalized; the species epithet is always Latinized and is never capitalized. The genus and species are in italics, followed by the name of the original describer of that species and the year in which it was named in standard uppercase. For example, Carnegiella marthae MEYERS 1927 tells us that this fish, the black-winged hatchetfish, was first described and named by Dr. George Meyers in 1927. When the describer and date are in parentheses, as for the Black Phantom Tetra Hyphessobrycon megalopterus (EIGENMANN, 1915), it indicates that the species is no longer in the genus to which this species was first assigned. In this example, Eigenmann originally placed this fish in the genus Megalamphodus but Stanley Weitzman and Lisa Palmer determined that the fish actually shares certain phylogenetic characteristics with the other species in the rosy tetra clade within the Hyphessobrycon genus, and in 1997 they published their findings and re-assigned the species to Hyphessobrycon. But as Eigenmann was the first to describe this fish as a new and distinct species, his choice of the species name remains valid and his own name as the describer is placed in parentheses.
The genus name quite often comes from some feature of the fish in that particular group; for example, the pencilfish are now all in the single genus Nannostomus, which comes from the Greek nanno (= small) and stomus (= mouth). When a scientist describing a new species recognizes that the fish has characteristics that are not common with the fish in all other existing genera, a new genus may be erected. The fish responsible for the establishment of a new genus is called the type species, which means that it has the special characteristics that will identify all future species within that genus.
The species epithet may denote some feature of the species, such as the false or green neon tetra, Paracheirodon simulans [= similar] in reference to the similarity in colour and pattern of this fish to Paracheirodon innesi [the neon tetra]; or it may honour the discoverer or another individual, as for example Hemigrammus bleheri named after Heiko Bleher who discovered the species; or it may refer to the location where it occurs, as for Corydoras guapore that inhabits the Rio Guapore system in South America.
The value of this binomial nomenclature system is certainty and clarity. Common names often differ even from one part of a country to another part, and certainly vary from one country to another, and are usually specific only to that language. In contrast, the scientific name can be used all over the world, in all languages, avoiding confusion and difficulties of translation. And it creates stability; though far from being absolute, the stability from initial name onwards is still an advantage for science and the hobbyist.