Learn Hardcore Japanese: tyuui ga hituyou desu.

注意 (chūi): a noun meaning “attention” or “caution.”
が (ga): the subject-marking particle, indicating that “caution” is what’s necessary.
必要 (hitsuyō): a na-adjective meaning “necessary.”
です (desu): the polite copula, turning “必要” into a polite predicate.

So literally it’s “Caution (が) necessary (です).”

Why is the particle が used after 注意 instead of は?

The pattern Xが必要だ/です uses が to present X as the thing whose existence is required.

が simply marks “caution” as the subject: “It is caution that is necessary.”
If you used は, you’d be making “caution” the topic and might imply contrast (e.g. “As for caution, it’s necessary—but something else may not be”).

What part of speech is 必要, and why does です follow it without another particle?

必要 is a na-adjective (形容動詞). When used predicatively:

In plain speech: 必要だ
In polite speech: 必要です
No extra particle is needed between a na-adjective and だ/です, just like 静かだ or 元気です.