
best
The highest comparative form (superlative) of the adjective 'gut' (good); expresses the highest quality or excellence.
Das war der beste Film, den ich je gesehen habe.
That was the best film I have ever seen.
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The highest comparative form (superlative) of the adjective 'gut' (good); expresses the highest quality or excellence.
Das war der beste Film, den ich je gesehen habe.
That was the best film I have ever seen.

best
Used as an adverb to express that something happens in the best way, or to give advice.
Am besten, du rufst ihn direkt an, um die Sache zu klären.
It would be best if you called him directly to clear up the matter.

to do one's best
A fixed expression meaning to try as hard as possible.
Auch wenn es schwer wird, werde ich mein Bestes geben.
Even if it gets difficult, I will do my best.
You have to distinguish between the form before a noun ('der beste Wein' — the best wine) and the adverbial or predicative form ('am besten' — best). A typical mistake is using 'am besten' before a noun.
Das ist der beste Kuchen, den ich je gegessen habe.
This is the best cake I have ever eaten.
To give advice, 'Am besten' (it would be best) can be placed at the beginning of the sentence. The verb then follows directly in position two.
Am besten, wir treffen uns direkt am Bahnhof.
It would be best if we met directly at the train station.
In the fixed expression 'sein Bestes geben' (to do one's best), the possessive pronoun adapts to the person who is making the effort. So it is 'ich gebe mein Bestes' (I do my best), 'du gibst dein Bestes' (you do your best), and so on.
Ich weiß, dass du dein Bestes geben wirst.
I know that you will do your best.
superior
„Best“ refers to the highest quality within a comparison and places something at the top. „Superior“ also means better or of higher quality, but it functions more strongly as a special quality category and is used, for example, for products or hotel rooms. Use „best“ when a ranking is meant and „superior“ when an upgraded category is being emphasized.

Von den drei angebotenen Zimmern ist dieses das beste.
Of the three rooms on offer, this one is the best.
Wir haben ein Superior-Zimmer mit mehr Platz und einem größeren Balkon gebucht.
We booked a superior room with more space and a larger balcony.