In zoological nomenclature, the specific name (also specific epithet or species epithet) is the second part (the second name) within the scientific name of a species (a binomen). The first part of the name of a species is the name of the genus or the generic name. The rules and regulations governing the giving of a new species name are explained in the article species description.
Historically, this was not the case. Carl Linnaeus, who formalized binomial nomenclature, made explicit distinctions between specific, generic, and trivial names. The generic name was that of the genus, the first in the binomial, the trivial name was the second name in the binomial, and the specific the proper term for the combination of the two. For example the binomial name of the tiger, Panthera tigris.
This was the proper usage from the 18th century into the late 20th century, although many authors seemed to be unaware of the distinctions between trivial and specific names and inconsistent and erroneous usage even appeared the International Code of Zoölogical Nomenclature. 
Grammatically, a binomen (and a trinomen, also) must be treated as if it were a Latin phrase, no matter which language the words were originally taken from. (This gives some justification to the popular usage of the phrase "Latin name" instead of the more correct phrase "scientific name".) Grammatically (using Latin grammar), the specific name must adhere to certain conventions. The specific name can be formed as:
In botanical nomenclature, "name" always refers to the whole name (of a species or otherwise), whereas in zoological nomenclature it can refer to either part of the binomen. Thus Hedera helix (common ivy, English ivy) is the name of the species; Hedera is the name of the genus; but helix is called the specific epithet, not the specific name.