[noun] + ではありません / じゃありません
[na-adj] + ではありません / じゃありません
[noun / na-adj] + ではありませんでした / じゃありませんでした
ではありません is the negative of です: "am not", "is not", "are not". Put it where です would be, after a noun or a な-adjective, and leave the remaining words as they are. The は in it is read wa.
It gives the no answer when someone asks what a person or thing is.
In speech では shortens to じゃ. ではありません is what you write and what you say on formal occasions. じゃありません is what you mostly hear in conversation, and it is still polite. じゃないです is a step more relaxed again.
ではありません does not go after an い-adjective or a verb. Each has a negative of its own: the adjective's い becomes くない, and the verb's ます becomes ません.
A な-adjective is negated the same way as a noun.
Add でした for "was not".
These go beyond the polite form.
In plain speech the same negative is ではない, or じゃない in conversation.