
der Täter
perpetrator
A person who has committed a crime or another unlawful act.
Die Polizei konnte den Täter kurz nach dem Einbruch festnehmen.
The police were able to arrest the perpetrator shortly after the burglary.
perpetrator / culprit
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perpetrator
A person who has committed a crime or another unlawful act.
Die Polizei konnte den Täter kurz nach dem Einbruch festnehmen.
The police were able to arrest the perpetrator shortly after the burglary.

culprit
More generally, a person who caused or carried out a particular, often negative, action.
Wer der Täter war, der die ganze Torte allein gegessen hat, blieb ein Geheimnis.
Who the culprit was who ate the whole cake alone remained a mystery.
The word 'Täter' (perpetrator/culprit) derives directly from 'die Tat' (an action or a crime). The ending '-er' denotes a person who does something. A 'Täter' is therefore literally a person who commits an act.
Jede kriminelle Tat hat einen Täter.
Every criminal act has a perpetrator.
In legal and police contexts, 'Täter' (perpetrator) is often used with certain verbs. Typical combinations are 'den Täter fassen' (to catch the perpetrator), 'den Täter ermitteln' (to identify the perpetrator), or 'den Täter überführen' (to prove the perpetrator's guilt).
Die Polizei hofft, den Täter bald zu fassen.
The police hope to catch the perpetrator soon.
In the language of the police, the justice system, and the media, 'Täter' (perpetrator) is the neutral and factual standard term for a person who has committed a crime.
Der Täter wurde zu einer Haftstrafe verurteilt.
The perpetrator was sentenced to prison.
Outside of a criminal context, 'Täter' (culprit) is often used humorously or ironically for someone who has committed a small, undesirable action.
Der Täter, der den letzten Keks gegessen hat, soll sich bitte melden!
The culprit who ate the last cookie should please come forward!
criminal
Both words can refer to a person after a serious crime. „Der Täter“ focuses primarily on the specific act and remains matter-of-fact. „Der Verbrecher“ refers to the person more generally and is clearly more judgmental, especially in the case of one or more serious crimes. „Täter“ is usually appropriate in a report, while „Verbrecher“ is more of a general judgment.

Die Polizei sucht den Täter, der gestern den Laden überfallen hat.
The police are looking for the perpetrator who robbed the store yesterday.
Nach mehreren schweren Überfällen gilt der Mann als gefährlicher Verbrecher.
After several serious robberies, the man is considered a dangerous criminal.