RANKS
The International Code of Botanical Nomenclature, which determines how plants are named, lists several different ranks, or levels of grouping for plants.  Bear in mind that the Code was designed by humans, for humans. It was also developed at a time when the concept of  a 'testable hypothesis' had not been developed and when there were fewer people involved in taxonomy (but I do not know if there were proportionately fewer people).  Now, back to the Code.  

Principal ranks: The Code states that every plant belongs to one, and only one, species and that every species belongs in a genus, every genus in a  family, every family, in an order, every order in a class, every class in a division (which may be referred to as a phylum), and every division in a kingdom.  These are the principal ranks, the ranks to which every species belongs. 

Secondary ranks: There are also several secondary ranks.  These are ranks that may be used, but need not be.   The secondary ranks are generally used to subdivide large groups.  Thus, a large family may be divided into tribes, a large genus into sections, large sections in series, large species into varieties, and large varieties into forms.   "Large" is used rather loosely here.  So long as there are two elements in the group, one can recognize a subgroup.  For instance, one could put the two genera of a single family into two tribes, each tribe having one genus, but this is not usually done. 

If that is not enough ranks, one can always create additional ranks immediately below any or all of the principal or secondary ranks by adding the prefix "sub-" to the rank concerned.  For instance, subspecies is a rank immediately below a species but above a variety.  Subfamily is a rank immediately below a family but above a tribe. Similarly, one can insert ranks above any of the recognized ranks, e.g., a superorder or superdivision.

The manner in which names are to be formed  are summarized in the table below, which shows the ranks in descending order, from the highest, most inclusive, group to the lowest, most exclusive, group. 

Limits of the Code: What the Code does not do is state how one is to determine what is the appropriate rank for a group or how to determine whether a group of plants merits formal recognition as a taxonomic group.  The Code does not, in other words, provide any guidance in making taxonomic decisions.  It merely states how to decide the name of a group that you have decided warrants formal taxonomic recognition.  Taxonomy comes first; nomenclature second. 

NAMES
In the table that follows, Red indicates the principal ranks.  The italicized name in the first column is the formal, Latin, name for the rank. The second column gives a quick summary of how names at that rank are to be formed.  The third column gives some examples.   In keeping with the practice adopted in the most recent issue of the Code, all scientific names are italicized. Only names above the rank of species begin with an upper case letter.

You absolutely must learn to recognize when a name is a family name, when it is probably a generic name, and when it is a species name (this last is easy - the name of a species is ALWAYS a binomial). You should also learn the endings of the principal ranks if you are wishing to impress me.

None of the nomenclatural codes recognize the rank of Domain. It is a rather recent rank, created after it was realized that bacteria fall into at least two very distinct groups.  So far as I am aware, taxonomists of all persuasions (bacterial, viral, and animal) accept it. 

 Alternative Family Names
There are eight family names that have been used for so long that they are accepted as correct, despite the fact that they do not follow the rules.  They are listed in the table below, together with the name that the family has if one insists on using family names that are based on a generic name.  I prefer the old names, but there is an increasing tendency is to to the regulated names.  Nice to feel like a dodo.

Warning: Fabaceae is sometimes interpreted to mean just the pea/bean portion of the Leguminosae; the meaning of Leguminosae always includes  the mimosoids and caesalpinoids as well as the pea and bean taxa. Similarly, Guttiferae always includes both the Clusiaceae and Hypericaceae (possibly with some other taxa as well). 

Authors of plant names, aka authorities
The meaning of a plant name is determined by the person(s) who first publishes it. This person (or these people) are known as the author(s) for the name. For some reason, the practice used to be to call these people the authority. 

The practice of citing an author with a scientific name started early on, long before the International Code was adapted. To give credit to the person involved, and also help other people understand what was meant by a name, taxonomists adopted the practice of writing the name of the person whose meaning they were following after the plant name, sometimes with a note as to where he (in the early days, always a 'he') described it.  For instance, Linnaeus noted that he included in what we would call Solanum tuberosum the species that Bauhin (1623) called Solanum pomiferum, fructo rotundo striato molli on page 167 of his Pinax theatri botanici and p. 89 of his Prodromus theatri botanici (Bauhin 1620).  

Origin of binomials. Consider another species, one that we would now call Solanum tuberosum, better known to you as potato. In Species Plantarum, Linnaeus called this species Solanum caule inermi herbaceo, foliis pinnatis integerrimus.   Freely translated, this means the Solanum with herbaceous, unarmed stems, pinnately compound leaves with entire margins.  Even Linnaeus found this a trifle long, so he wrote in the margin 'tuberosum'.  He considered 'Solanum tuberosum' to be the trivial name of the plant, the equivalent of what we would now call a common name, but a common name understood by anyone in the world with grade school education (Latin was taught in grade school). For another example, see the discussion of Zea in Species plantarum.

Taxonomists before Linnaeus had suggested using binomial names for species; Linnaeus did not agree, but by providing a marginal name for all the many species he described, he provided the impetus that led to widespread agreement that binomials are better than long phrases.  When the International Code of Botanical Nomenclature was first proposed, it was formally agreed that vascular plant nomenclature would start with Linnaeus' trivial names.  (Later it was decided that the starting point for some groups, notably fungi and algae, would be other works, but that does not concern us). 

If you wish to make it clear that you are using Linnaeus' concept of Solanum tuberosum, you need to place his name, or  the abbreviation 'L.' after Solanum tuberosum, i.e., Solanum tuberosum L.

Authors, part 2
OK, so you appreciate that the point of citing an author for a scientific name is to let others know who first described it.  Why do this? So that you can look at the original description and, better yet, the specimens that they examined. It is not a matter of giving honor.  (There is more to it than this, but that is enough for now).   Consider the species commonly known in Utah as 'Serviceberry' (but in Canada as Saskatoon Berry).   Its scientific name is Amelanchier alnifolia, but the authorities for the name are given as (Nuttall) Nuttall ex M. Roemer?  Why are there three author names listed?

1. The first Nuttall, the one in parentheses indicates that Thomas Nuttall was the first person to recognize the plants that we now call Amelanchier alnifolia as distinctive, but he put them in a different genus, the genus Aronia. So the first name given to these plants was Aronia alnifolia. Unfortunately, Nuttall was unaware that the name Aronia was first used for a group of plants that have nothing to do with the genus that we now call Amelanchier. Max Roemer was preparing a publication in which the alnifolia plants had to be treated. Nuttall told him that they should be placed in the genus Ameleanchier. "Nuttall ex M. Roemer" tells the initiated that it is Roemer's name that will be found on the publication but, being a genetleman and a scholar, he acknowledged that it was Nuttall that told him a new name was needed. If Nuttall had published the paper, the authorship of Amelanchier alnifolia would be (Nuttall) Nuttall.

A note for zoologists: In zoology, it is common only to list the original author, the one in parentheses. This is unacceptable in botany.