
die Haft
imprisonment
The state of being in prison; a prison sentence.
Der Angeklagte wurde zu drei Jahren Haft verurteilt.
The defendant was sentenced to three years' imprisonment.
custody / imprisonment
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imprisonment
The state of being in prison; a prison sentence.
Der Angeklagte wurde zu drei Jahren Haft verurteilt.
The defendant was sentenced to three years' imprisonment.

in custody
The situation in which a person is being held by the police or judicial authorities, often before a verdict, as in pre-trial detention.
Der Verdächtige befindet sich seit gestern in Haft.
The suspect has been in custody since yesterday.
'Haft' (custody/imprisonment) belongs to the word family of 'haften' in the sense of 'festhalten' (to hold onto/detain). The basic idea is that a person is being held.
Der Verdächtige wurde in Haft genommen.
The suspect was taken into custody.
The word 'Haft' (custody/imprisonment) is a formal term from legal and official language. It is mainly used in official contexts such as news reports, court reports, or police statements.
Die Staatsanwaltschaft forderte eine lange Haftstrafe.
The public prosecutor demanded a long prison sentence.
When referring to a prison sentence, typical phrases are: 'zu Haft verurteilt werden' (to be sentenced to imprisonment), 'eine Haftstrafe verbüßen' (to serve a prison sentence), or 'aus der Haft entlassen werden' (to be released from prison).
Er wurde zu fünf Jahren Haft ohne Bewährung verurteilt.
He was sentenced to five years' imprisonment without parole.
The phrase 'in Haft' (in custody) is a fixed combination and describes the state of being detained. Typical verbs are 'in Haft sein/sitzen' (to be in custody), 'sich in Haft befinden' (to be in custody), or 'jemanden in Haft nehmen' (to take someone into custody).
Der Mann sitzt seit zwei Wochen in Untersuchungshaft.
The man has been in pre-trial detention for two weeks.
prison
Both words can refer to the situation of serving a prison sentence. Gefängnis mainly refers to the building or institution, while Haft describes the state of being imprisoned. When the focus is on the place, use Gefängnis; when it is on the duration or state, use Haft.

Er muss drei Jahre in Haft bleiben.
He has to remain in custody for three years.
Er muss drei Jahre im Gefängnis bleiben.
He has to remain in prison for three years.