sich scheiden lassen
to get divorced
To officially end a marriage in court.
Nach vielen Jahren haben sie beschlossen, sich scheiden zu lassen.
After many years, they decided to get divorced.

to divorce / separate
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to get divorced
To officially end a marriage in court.
Nach vielen Jahren haben sie beschlossen, sich scheiden zu lassen.
After many years, they decided to get divorced.

The verb 'scheiden' carries the basic meaning of 'trennen' (to separate). You can also find this idea in related words such as 'entscheiden' (to decide; literally, to separate one choice from another), 'unterscheiden' (to distinguish), or 'der Abschied' (farewell; the separation from someone).
Es ist schwer, zwischen den beiden fast identischen Produkten zu unterscheiden.
It is difficult to distinguish between the two almost identical products.
The perfect tense is usually formed with 'haben' (to have). The auxiliary verb 'sein' (to be) is used only in the archaic or poetic meaning 'weggehen' (to go away) or 'sterben' (to die).
Das Paar hat sich letztes Jahr scheiden lassen.
The couple got divorced last year.
When talking about a divorce, you use the fixed structure 'sich scheiden lassen von' followed by a person in the dative. The verb 'lassen' (to let/have something done) indicates that it is an official process.
Sie hat sich von ihrem Mann scheiden lassen.
She got divorced from her husband.
In the meaning 'trennen' (to separate), 'scheiden' is used mainly in technical, chemical, or very formal contexts. A well-known idiom is 'die Spreu vom Weizen scheiden' (to separate the wheat from the chaff).
Im Labor wird die Flüssigkeit in ihre Bestandteile geschieden.
In the laboratory, the liquid is separated into its components.
The idiom 'hier scheiden sich die Geister' is very common. It means that there are very different and incompatible opinions on a topic.
Sollte man früh aufstehen oder lange schlafen? Daran scheiden sich die Geister.
Should you get up early or sleep late? Opinions are divided on that.
The meaning 'verlassen' (to leave) or 'weggehen' (to go away) is archaic today and is used almost only in literature, in poems, or in fixed elevated expressions like 'aus dem Leben scheiden' (to die).
Der alte König schied friedlich aus dem Leben.
The old king peacefully departed from life.
to leave / rely
'verlassen' and 'scheiden' can both express that someone leaves a place or a person. The important difference is style. 'verlassen' is the normal everyday word and fits in almost all situations. 'scheiden' in this meaning sounds old-fashioned, solemn, or poetic today. You are more likely to find it in songs, poems, or very elevated language. In normal conversation, people almost always say 'verlassen'.

Schweren Herzens schied der alte Kapitän von seinem Schiff.
With a heavy heart, the old captain parted from his ship.
Vor der Reise verlässt er das Haus schon um sechs Uhr.
Before the trip, he leaves the house at six o'clock.