1First in a sequence or ranking — the 'number-one' or 'leading' position. Used in expressions like 第一の候補 ('top candidate'), 第一志望 ('first-choice school'), 第一印象 ('first impression'), 第一回 ('the inaugural session'). Most commonly attaches to a following noun with the particle の.
面接では第一印象がとても大事だ。
In a job interview, the first impression is extremely important.
第一志望の大学に合格して、嬉しくて泣いた。
I got into my first-choice university and cried with joy.
次期社長の第一候補として、山田さんの名前が挙がっている。
Yamada is being floated as the top candidate for the next president.
2Most important, top priority. Used predicatively as in 家族が第一 ('family comes first') or 安全第一 ('safety first'), or adverbially in 第一に〜 to introduce one's top priority or main reason. An extension of sense 1: not just 'first in sequence' but 'first in importance'.
何より健康が第一だよ。
More than anything, your health comes first.
工事現場では安全第一を徹底してください。
At a construction site, please make safety-first your top rule.
第一に考えるべきは、お客様の満足だ。
The number-one thing to consider is the customer's satisfaction.
3The best or greatest within a given domain or area. Functions as a superlative: 業界第一のシェア ('the top market share in the industry'), 日本第一の高さ ('the greatest height in Japan'). Slightly written-register and promotional in feel.
この会社は業界第一のシェアを誇っている。
This company boasts the top market share in the industry.
東京スカイツリーは日本第一の高さを誇る電波塔だ。
Tokyo Skytree is the tallest broadcasting tower in Japan.
4Used adverbially at the start of a sentence to introduce a primary reason or fundamental point — 'first of all', 'to begin with', 'in any case'. In conversation, often used to push back on someone's view, similar in tone to だいたい or そもそも: 第一、〜 ('To start with, ...').
第一、そんな計画は時間的に無理だよ。
To start with, a plan like that just isn't feasible time-wise.
第一、話を最後まで聞いてから判断してください。
First of all, please hear me out before judging.
「行きたくない。第一、お金がないし、時間もない。」
'I don't want to go. For one thing, I don't have the money, and I don't have the time either.'