じゃない is the casual form of ではない. では is regularly contracted to じゃ in Japanese, so is worth remembering as its own structure. じゃない (casual), or ではない (semi-polite) are the structures used for creating the negative form of nouns and な-Adjectives.

These structures are simply added to the base form of the noun or な-Adjective.

じゃ and では can also be used with the negative ます form of ある, to create the polite variant.

ない is sometimes said/written as ね or ねえ. However, this is considered very casual, or even a bit rough. Accordingly, it is best avoided in most conversations.

じゃなかった is the casual form of ではなかった. では itself is regularly contracted to じゃ in Japanese, so is worth remembering as its own structure. じゃなかった (casual), or ではなかった (semi-polite) are the structures used for creating the negative-past form of nouns and な-Adjectives.

Either じゃなかった or ではなかった will simply be added to the base form of a noun or な-Adjective.

じゃ and では may also be used with the negative-past ます form of ある, to create the polite variant.

In order to create the negative past form of い-Adjectives, we will need to use the conjugation form. The conjugation form of an い-Adjective is simply created by removing the い, and replacing it with く. く is then followed by ない (conjugated into its past form なかった), or あります (conjugated into its past negative form ありませんでした).

The polite negative (past) form requires use of the う - Verb, ある. This is not a unique rule for the negative past, but actually the correct way that all く-form conjugations of い-Adjectives should be constructed. However, in modern Japanese, it is not considered necessary. Instead, the semi-polite form is often used.

The semi-polite form is constructed by using なかった, and then simply adding です.