で Copula

In Japanese, the de で particle is sometimes actually a copula. Specifically, it's the te て form of the da だ copula, and consequently the te て form of na-adjectives and no-adjectives.

Grammar
- Compounds

Grammar

The te て form can be used to connect two adjectives.

chiisai
小さい
[It] is small.
kawaii
可愛い
[It] is cute.
chiisakute kawaii
小さくて可愛い
[It] is small and, is cute.

Above, we see the te form of i-adjectives, ~kute ～くて.

With na-adjectives, the na な copula is used attributively, and the de で copula is used as the te て form.

heibon da
平凡だ
[It] is average.
{heibon na} koukousei
平凡な高校生
A high school student [who] {is average}.
An {average} high school student.
{heibon de tsumaranai} koukousei
平凡でつまらない高校生
A high school student [who] {is average and, is boring}.
An {average and boring} high school student.

With no-adjectives, the no の copula is used attributively, and the de で copula is used as the te て form.

futsuu da
普通だ
[It] is normal. (noun.)
{futsuu no} koukousei
普通の高校生
A high school student [who] {is normal}.
A {normal} high school student.
{futsuu de tsumaranai} koukousei
普通でつまらない高校生
A high school student [who] {is normal and, is boring}.
A {normal and boring} high school student.

The de で copula can connect a na-adjective to a no-adjective, or vice-versa, too.

heibon de futsuu da
平凡で普通だ
[It] is average and, is normal.
futsuu de heibon da
普通で平凡だ
[It] is normal and, average.

In the attributive, the copula of the last adjective, right before the noun, is used to connect it to the noun.

{heibon de futsuu no} koukousei
平凡で普通の高校生
A high school student [who] {is average and, is normal}.
- Here, futsuu, a no-adjective, comes before the noun koukousei, so the no の copula is used.
{futsuu de heibon na} koukousei
普通で平凡な高校生
A high school student [who] {is normal and, is average}.
- Here, the na-adjective comes before the noun, so the na な copula is used instead.

Compounds

Like any other te て form, the de で copula can be marked by the wa は particle, forming the compound dewa では.

futsuu dewa nai
普通ではない
Normal, it isn't.
futsuu dewa aru
普通ではある
Normal, it is.

This dewa では compound is often contracted to ja じゃ, sometimes to jaa じゃあ.

In particular, dewa nai ではない, or janai じゃない, is the normal way to say something "is not" in Japanese.

sono koukousei wa futsuu janai!
その高校生は普通じゃない！
That high school student isn't normal!

The phrases sore dewa それでは and sore ja それじゃ are often used to say "bye."" and so are just dewa では and jaa じゃあ.

Like other te-wa ては phrases, it sometimes forms a condition, whose consequence is often omitted.