In botanical taxonomy, the binomial is the fundamental unit, consisting of a genus name
and a specific epithet. The first letter of the genus is capitalized, the specific epithet is
all lowercase. Both are given in italics or, more commonly in handwriting, underlined.
For instance, in Krascheninnikovia lanata, Krascheninnikovia is the genus name and
lanata is the specific epithet. In technical writing on taxonomy, the binomial is
followed by one or more authorities. These are the people responsible for formally naming
the plant in question, with their names given in a standard, usually abbreviated, form.
Authorities are not italicized. If a species has moved from one genus to another over time, the
author(s) to name the species are given in parentheses, and the author(s) to place it in
its current genus are given after, without parentheses. For instance, we have
Krascheninnikovia lanata (Pursh) A.Meeuse & A.Smit. Frederick T. Pursh named the
species as Diotis lanata, Meeuse and Smit then later placed it in the genus
Although the binomial is the basic unit, we can also have trinomials, in which taxa below the species level are named. One difference between botanical and zoological nomenclature is that botany has two levels below the species: subspecies and variety. Although it is technically possible to create tetranomials (e.g., X y ssp. y' var. y'') this is now rare. Normally only trinomials are used (e.g., X y ssp. y' or X y var. y') and the distinction between the variety and subspecies has become, for the most part, a formal detail of little real importance. Authorities for subspecific or varietal names are listed separately, but subspecific and varietal names are treated as equivalent to specific epithets in terms of tracing authority. As an example, in the name Agave parryi Engelm. ssp. neomexicana (Woot. & Standl.) B.Ullrich, first Engelmann named Agave parryi and Wooton and Standley named Agave neomexicana, and later Ullrich moved Agave neomexicana to subspecific status within Agave parryi. Wooton and Standley remain the original authors of Agave parryi ssp. neomexicana despite the change in rank and placement of the taxon.
Right now everything on this site is organized by binomials. I omit authorities, however, both for the sake of brevity and because they are rarely important in understanding a name (problems only arise when different authors have inadvertently used the same name for different plants). I use binomials for several reasons. First, the systematic names are how I usually think about things, so I know them and don't have to sit around looking things up all the time. Second, a lot of species have multiple common names, or no common name, or different common names in different places, etc. Further, a single common name can be used for multiple species. So in organizing things according to common names, you either have to do a lot of cross-listing and leave out some species, or make up some names for nameless taxa and arbitrarily choose one of the common names to use for taxa with many names. Both approaches are problematic. Furthermore, common names rarely say much about the relationships between plants, so they are generally either uninformative or misinformative when it comes to trying to understand how plants are related.
That said, coming up with alternative common name listings for things is somewhere on my list of things to do, but not very high on the list and a time-consuming enough task I don't know if it'll happen.