
absitzen
to serve
To serve a prison sentence.
Er musste eine Haftstrafe von drei Jahren absitzen.
He had to serve a three-year prison sentence.
to serve / to sit out / to dismount
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to serve
To serve a prison sentence.
Er musste eine Haftstrafe von drei Jahren absitzen.
He had to serve a three-year prison sentence.

to sit out
To let a certain period of time pass while doing nothing or being bored until it is over.
Wir mussten die zwei Stunden Verspätung am Bahnhof absitzen.
We had to sit out the two-hour delay at the station.

to dismount
To get off a riding animal, such as a horse, or a vehicle, such as a bicycle.
Nach dem langen Ritt waren die Reiter froh, endlich absitzen zu können.
After the long ride, the riders were glad finally to be able to dismount.
'Absitzen' is a separable verb. The prefix 'ab' goes to the end in a main clause. In subordinate clauses, the verb stays together, and in the infinitive with 'zu', 'zu' goes in the middle.
Er saß seine Strafe bis zum letzten Tag ab.
He served his sentence until the very last day.
The object of the sentence reveals the meaning. 'Eine Strafe' (a sentence/punishment) or 'Jahre' (years) with 'absitzen' means serving a prison sentence. 'Die Zeit' (the time) or 'Stunden' (hours) with 'absitzen' means waiting in a bored way. Without an object or with 'von' (e.g. 'vom Pferd'—from the horse), it means to dismount.
Er musste zwei Jahre im Gefängnis absitzen.
He had to serve two years in prison.
In the meaning 'to serve a sentence', the verb is almost always used with an accusative object that names the type or length of the sentence. Typical phrases are 'eine Haftstrafe absitzen' (to serve a prison sentence) or 'eine Gefängnisstrafe absitzen' (to serve a jail/prison sentence).
Wegen des Betrugs musste der Manager eine lange Haftstrafe absitzen.
Because of the fraud, the manager had to serve a long prison sentence.
When 'absitzen' means 'absteigen' (to dismount / to get off), it is often used with the preposition 'von' and the dative to say what someone is getting off. But it can also stand without this addition.
Die Radfahrer saßen von ihren Rädern ab, um eine Pause zu machen.
The cyclists got off their bikes to take a break.
to dismount / descend
Absteigen and absitzen can both refer to getting down from a horse. Absteigen is the general choice and also works for bicycles or mountains. Absitzen is used especially when riding and in technical or military contexts.

Auf den Befehl des Reitlehrers sitzt die Reiterin sofort vom Pferd ab.
At the riding instructor's command, the rider immediately dismounts.
Vor dem geschlossenen Tor steigt die Reiterin kurz vom Pferd ab.
The rider briefly dismounts from her horse in front of the closed gate.