[noun] + だ [na-adj] + だ [noun] / [na-adj] + じゃない / ではない [noun] / [na-adj] + だった [noun] / [na-adj] + じゃなかった / ではなかった
だ is the plain form of です: "is", after a noun or a な-adjective. You use it with friends and family, and in writing that does not address the reader politely – diaries, novels, essays.
だ changes the same way です does, and the noun or adjective in front never changes. ではない is the written and more careful form of じゃない.
A な-adjective takes な when it comes before a noun. At the end of a sentence that な becomes だ, or です when you are polite – never なです.
An い-adjective already means "is …", and a verb is complete on its own. Neither takes だ. With an い-adjective, です adds politeness only, and the plain form is the adjective by itself.
In casual speech, a bare だ at the end of a sentence sounds blunt. It is often left off, or softened with よ or ね; before ね it can drop as well, and きれいね is softer than きれいだね. A casual question drops it altogether and rises at the end, as in がくせい学生きごう? "a student?".
Inside a sentence, before けど, から or とおもう思う, だ stays.