It is a suffix. It is formed in this way "verb + べき", but "する" is an exception and you can say "するべき(colloquial style)" or "すべき(literary style)". It means "should do" and it is used when giving a suggestion or advice from a common sense perspective.
"した方がいい" is a similar expression, but it sounds less obligatory than "べき".
Ken is making plans to go to Tokyo on vacation.
- Ken: Where should I go in Tokyo?
東京で、どこに行くべきかな？ - Leo: Well, if you're interested in anime, you have to go to Akihabara!
アニメが好きなら、秋葉原に行くべきだよ。 - Ken: Great.
いいね。 - Leo: By the way, you still have a lot of stuff to do before your vacation.
ところでさ、休暇前に、まだやるべき事が沢山あるよ。
Ken is assembling a desk by himself.
- Ken: I want to compose the TV stand, but I can't.
TV台を組み立てたいんだけど、できない。 - Leo: Have you read the instruction manual?
説明書を読んだ？ - Ken: No.
ううん。 - Leo: Look. The important thing that you should know is written here.
見て。知っておくべき、重要な事は、ここに書かれているよ。
Meg's mom is angry because Meg was out all night without telling her anything and got home early the next morning.
- Meg: I'm home!
ただいま！ - Mom: Why didn't you call me if you planned to stay overnight? I thought something had happened to you.
外泊するなら、なんで連絡しなかったの？何か起こったのかと思って、心配したよ。 - Meg: Mom, don't worry about me. I'm not a child!
心配しないでよ。子供じゃないんだから。 - Mom: I know that, but you have to let me know if you're going to be out of all night.
分かっているけど、事前に連絡するべきよ。
Ken is worried about Leo because he doesn't look well.
- Ken: You've been coughing all day. You should go home early and rest.
ずっと咳をしているね。早く帰って、休んだ方がいいんじゃない？ - Leo: Hmm. I have to prepare for the conference call in two days.
んー、明後日の会議の準備をしなきゃ。 - Ken: You can do it tomorrow and I'll help you. You'd better rest. There's no point in getting ready for the conference if you 're not even going to be able to go.
明日できるし、僕が手伝うよ。休むべきだよ。会議に出れなかったら、意味がないだろう。
You can use べき to talk about something you shold do too, like the following. Usually, you need to use a passive verb.
- Ken: When will I have to submit the documents?
いつ資料を提出しなきゃいけないの？ - Leo: They are supposed to use at the meeting in two days. So I think they should be submitted by tomorrow.
資料は二日後の会議で使われるはずだから、明日までに提出されるべきだ。
The test for coronavirus should be done for people who have symptoms such as fever and cough.
熱や咳などの症状がある人に対して、コロナの検査は実施されるべきだ。
熱や咳などの症状がある人に対して、コロナの検査は実施されるべきだ。