A botanical name is a formal scientific name which follows the International Code of Nomenclature for algae, fungi, and plants (ICN). If it concerns a plant cultigen, the additional cultivar term must follow the International Code of Nomenclature for Cultivated Plants. 

The code of nomenclature covers "all organisms traditionally treated as plants, whether fossil or non-fossil, e.g. blue-green algae (Cyanobacteria); fungi, including chytrids, oomycetes, and slime moulds; photosynthetic protists and taxonomically related non-photosynthetic groups".

The purpose of a formal name is to have a single name which is accepted and used worldwide for a particular plant or plant group. For example, the botanical name Bellis perennis was given to a plant species which is native to most of the countries of Europe and the Middle East, where it has various names in many languages. Later it has been introduced worldwide, bringing it into contact with more languages.

Some English names for this plant species are: daisy, English daisy, lawndaisy. In other languages, some examples are: , , , .

The cultivar Bellis perennis 'Aucubifolia' is a golden-variegated horticultural selection of this species.

Parts in a name 
Depending on rank, botanical names may be in one part (genus and above), two parts (species and above, but below the rank of genus) or three parts (below the rank of species):

in one part
Plantae (the plants)
Marchantiophyta (the liverworts)
Pinophyta (the conifers)
Magnoliopsida (class including the family Magnoliaceae)
Asteridae (subclass including the family Asteraceae)
Rosaceae (the rose family)
Fragaria (the strawberry genus)

in two parts
Acacia subg. Phyllodineae (the wattles)
Citrus sinensis (sweet orange)

in three parts
Calystegia sepium subsp. americana (American hedge bindweed)
Crataegus azarolus var. pontica (a Mediterranean hawthorn)

A name in three parts, i.e., an infraspecific name (a name for a taxon below the rank of species), needs a "connecting term" to indicate rank. In the Calystegia example above, this is "subsp." (for subspecies). In botany there are many ranks below that of species (in zoology there is only one such rank, subspecies, so that this "connecting term" is unnecessary there). A name of a "subdivision of a genus" also needs a connecting term (in the Acacia example above, this is "subg.", subgenus). The connecting term is not part of the name itself.

Binary name
For botanical nomenclature, the ICBN prescribes a two-part name or binary name for any taxon below the rank of genus down to, and including the rank of species. Taxa below the rank of species get a three part name. 

A binary name consists of the name of a genus and an epithet.

 In the case of a species this is a specific epithet:
Bellis perennis is the name of a species, with perennis the specific epithet. There is no connecting term involved, to indicate the rank

 In the case of a subdivision of a genus (subgenus, section, subsection, series, subseries, etc.) the name consists of the name of a genus and a subdivisional epithet. A connecting term should be placed before the subdivisional epithet to indicate the rank. 
Paraserianthes sect. Falcataria

More than two parts
 In the case of a cultivar there is an additional cultivar epithet (this is a non-Latin part of the botanical name and is not written in italics). The cultivar epithet may follow either the botanical name of the species, or the name of the genus only, or the common name of the genus or species (provided the common name is unambiguous). The generic name, followed by the cultivar name, is often used when the parentage of a particular hybrid cultivar is uncertain or when it cannot be linked with certainty to a particular species.
In Bellis perennis 'Aucubifolia', Bellis perennis is the name of the species, with perennis the specific epithet and 'Aucubifolia' the cultivar epithet. 
 
 Sometimes a subdivision of a genus may be indicated with a listing in three or more parts. However, this is not its formal name. Its botanical name is in two parts.

Related pages
 Biological classification
 Binomial nomenclature
 Taxonomy
 Systematics
 Cultigen

References

Other websites
 The International Code of Nomenclature for Cultivated Plants  

Plant taxonomy