べき expresses a similar meaning, but is much more forceful—to the point where it is almost an order. While you may hear it in speech, it is a considerably strong expression often used to indicate social, moral, or logical responsibilities. It is highly common in writing.
タバコをやめるべきです。
You should (must) quit smoking.
約束を守るべきだ。
You should keep promises (it’s your moral duty).
足が痛いなら医者に診てもらうべきだと思います。
If your leg hurts, I think you should have the doctor take a look at it.
Abbreviating する
When used with the verb する, the grammar is usually abbreviated from するべき to すべき.
留学すべきかどうかについて、親と真剣に相談します。
Seriously discuss with your parents whether or not you should study abroad.
Modifying Nouns & Idioms
Unlike はず, no particle is required to modify a noun. べき attaches directly to the noun it describes.
絶対守るべき約束。
A promise you should definitely keep.
べき can also be used with certain verbs to mean “worthy of” or “deserving of”.
驚くべき発見。
A surprising discovery (a discovery worthy of surprise).
The Negative (べきではない)
To say you “should not” do something, you conjugate the auxiliary verb (だ) into the negative (ではない). Critical JLPT Rule: You cannot attach べき to the negative form of the verb itself. For example, 渡らないべき is grammatically incorrect. It must always be 渡るべきではない.
車がなくても赤信号の時に、渡るべきではない。
Even if there aren’t any cars, you shouldn’t cross the road when the lights are red.
この町の治安が悪いから夜に家を出るべきではない。
This town isn’t safe and so at night you shouldn’t leave the house.
Literary/Archaic Forms (べからず / べし)
A formal, literary way of forming a strict negative command is with べからず. This is an old expression frequently seen on stern warning signs.
イギリスでは高速道路で時速７０マイル以上運転すべからず。
In England you should not drive at over 70mph on the motorways.
Incidentally, べし is the archaic dictionary form of べき. You may occasionally see it in formal writing or proverbs, but it is rarely ever used in modern speech.