Okay, this is my first question on this page so bear with me, I was studying Japanese nouns but I'm not sure if you use ではありません or じゃない to deny a noun, so if someone could please explain it to me, that would be great. Thanks!
Either is acceptable. Are you familiar with the distinction between the plain form and the polite form? The polite form is something you use with strangers and superiors, where the plain form would be used with family and friends. (The lines do blur slightly: some people might use polite forms with their parents and older sibling, but not their younger siblings.)
Polite forms end in ～ます. Negated, ～ます becomes ～ません.
You might know that a form of the copula is ～である. ある negated becomes ない, and a fossilized は is added; ～である becomes ～ではない. ～では is then contracted, becoming ～じゃない.
In the case of ～ではありません, ～である is first changed to polite ～であります, あります becomes ありません, and the same fossilized は is added. You now have ～ではありません. Note that ～じゃありません also exists, but it's somewhat rare; in more casual speech, people most frequently make ～じゃない into ～じゃないです, because it's somewhat like an adjective.
Please ask if any part of my answer requires explanation.
じゃ is a colloquial contraction of では, ない is the plain form of ありません so both are used but without the same level of politeness.
To clarify that a bit (I know it wasn't clear for me when I started Japanese) here is some information about the copula:
- だ, the plain form of the copula is a contraction of である (で+ある).
- You can think of です as the contraction of であります (で+あります) though it's not etymologicaly true from what I know.
- は in the negative versions is the contrastive particle は which gives the idea in brackets in: "it's not X (but something else)".
- ない is the negative version of ある.
So we have (plain on the left, polite on the right):
だ（である） です（であります） ではない ではありません だった でした（でありました） ではなかった ではありませんでした