The formal start of the nomenclature system in use today occurred in 1735 with the publication of Systema Naturae by Carl Linnaeus. He is correctly referred to as the father of modern taxonomy because the basic foundation he established has never changed. The grouping and ranking of organisms by species, genus, class, order and kingdom is essentially intact although there are always continuing (some controversial) revisions. This ranked based system of nomenclature/taxonomy is referred to as the Linnaean or Binomial or Scientific system. The “Binomial “ name comes from the practice of giving each type of organism two names, genus and species. The species name is also known as the “specific epithet,” and when added to the genus name forms the binomial name which sets one plant apart from all other plants, including any other plant which happens to have the same species name and any other plant in the same genus. Later even more specific identifications have been added, e.g., subspecies, variety, cultivar, but the binomial name remains at the heart of the Linnaean system nomenclature.

Linnaeus continued to develop and extend his classifications. In 1753 he published the first edition of Species Plantarum, a two volume, 1242 page work which included all the plant species known by Linnaeus. This work naturally gave much more attention to angiosperms (which now has about 350,000 species compared to about 960 gymnosperm species) which were grouped by their flower morphology, especially the number of stamens and pistol forms.
The rules governing the use of scientific nomenclature are encoded in Nomenclature Codes. There is one for the Animal Kingdom (the International Code of Zoological Nomenclature, or ICZN) and one for plants and everything else (the International Code of Botanic Nomenclature, or ICBN). These two codes vary in regards to some of the rules which must be followed when using the scientific names. 
The principal differences are that the I.C.B.N.:
(a) has its starting point in 1753, with the first published Species Plantarum (the I.C.Z.N. began in 1758, with the 10th edition of Systema Naturae).
(b) does not allow the use of genus/species tautonyms, i.e., the genus and species names cannot be the same.
(c) does not require the date of description if the author’s name is given (the I.C.Z.N. requires that the date follow the author’s name).
(d) requires that the abbreviation subsp. be used before the subspecies name and var. before the variety name.
(e) requires that the author of a no longer valid name for that species be enclosed in parentheses and in the case of well-known biologists, there is a list of standard abbreviations (which can be found in Wikipedia http://species.wikimedia.org/wiki/catalog.Taxon_Authorities/L ). For instance, the standard abbreviation for Linnaeus is L., for Philip Miller it is Mill., for E.A. Carriere it is Carriere, for David Don it is D. Don and for David Douglas it is Douglas.
Examples that illustrate these differences:
I.C.Z.N.:
Gorilla gorilla gorilla Linnaeus
Panthera tigris (Linnaeus, 1758) (the original name by Linnaeus in Systema Naturae was Felix tigris)
I.C.B.N.:
Pinus pinea L. (why did Linnaeus decide not to call it Pinus pinus? I don’t know.)
Larix decidua (L., 1753) Miller (Linnaeus’ original name in Species Plantarum was Pinus decidua)
Pinus ponderosa subsp. ponderosa
There is also an additional cultivated plant code under the I.C.B.N., the I.C.N.C.P., which regulates the naming of cultivars, hybrids and Groups. There are also special provisions for smaller photosynthetic organisms, e.g., cyanobacteria, fungi, fossils, etc. There is a separate code for bacteriology, the I.C.N.B., which began in 1980.
The I.C.B.N. and its subcodes are updated every five years when the International Botanical Congress meets (the next meeting is in Melbourne in 2011). The preceding meeting was in 2006 in Vienna and the final regulations at that time were referred to as the Vienna Code. Before that there was the 2001 meeting in St. Louis and the St Louis Code, etc.
Is there more? Yes! Today it is generally accepted that a structured classification of organisms by morphological similarities and differences is determined mostly by their evolutionary development. In 1753, Linnaeus had no concept of evolution; he was merely categorizing God’s creation so that it could be better understood by man. In 1859, with Darwin’s publication of The Origin of Species, it progressively became understood that taxonomy and phylogeny should be related, i.e., that the taxonomic organizational ranking should reflect evolutionary descent. This idea was given impetus by Willi Helwig’s concept of “phylogenetic systematics” in 1950 with the use of tree-like cladistic diagrams (clados = Gr. Branch). Soon after came two developments that accelerated the move to a phylogeny based taxonomy: (1) the use of polymerase chain reaction techniques that revealed the base sequences of the nucleic acids of nuclei, chloroplasts, etc. and (2) the development of computers which have software that can organize massive amounts of data well beyond the capacity of the human brain. Further information about cladistics and taxonomy may be found on this website under the heading ”Phylogenetic/Cladistic Taxonomy”.