じゃないです is a colloquial, uneducated way of saying
じゃありません. It is not totally ungrammatical, but is not totally correct either.
My reasoning for this is because it is not the shortest way of saying it. Assuming that
ない results from obligatory deletion of
ara in Tokyo dialect and that the
i-ending of an i-adjective is an obligatorily replaced form of
ku aru, the fully expanded form of the two expressions are:
ないです (7 morphemes)
ar-ana-ku ar-u des-u
DUMMYVERB-NEG-ADJECTIVE DUMMYVERB-NONPAST POLITE-NONPAST
ありません (4 morphemes)
ありません is much shorter. For those people who do not use
じゃありません, the negative morpheme
en, which is conditioned to be used only with politeness, is either blocked temporarily or is not known by the speaker.
I am interested in hearing other accounts of this contrast.
To answer ento's question:
In general (in any language), a well phrased (correct) sentence has to be the shortest among the alternative strategies that have the exact same meaning (including connotation, naunce, etc.). This is a genaral, basic fact about language, and is observed at various levels (word, sentence, discourse). For example, in English, it is not fully appropriate, unless there is a special connotation or purpose, to say sentences like
I am doing eating of a sandwich
I am feeling sadness
It is the case that it is sunny today [As a main sentence]
because there are shorter ways of saying them with the same meaning:
I am eating a sandwich
I am sad
It is sunny today
My explanation above handles
ありません in relation to this basic principle.
This is about awkardness, appropriateness, consiceness, and correctness, but is not directly related to formality.
As for the measure of shortness, there can be several possibilities, but for symplicity, I used the number of morphemes.