
der Fehler
mistake
An action, decision, or statement that is not correct or leads to an undesired result.
Ich habe einen Fehler in der Rechnung gemacht.
I made a mistake in the calculation.
mistake / error
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mistake
An action, decision, or statement that is not correct or leads to an undesired result.
Ich habe einen Fehler in der Rechnung gemacht.
I made a mistake in the calculation.

error / defect
A flaw, defect, or imperfection in an object, a device, or a system.
Die Waschmaschine hat einen technischen Fehler und funktioniert nicht.
The washing machine has a technical error and isn't working.

flaw
A negative character trait or weakness in a person.
Sein größter Fehler ist seine Ungeduld.
His biggest flaw is his impatience.
The most common and natural way to express that you do something wrong is the fixed combination 'einen Fehler machen' (to make a mistake). The more formal variant is 'einen Fehler begehen' (to commit a mistake/error).
Keine Sorge, jeder macht mal einen Fehler.
Don't worry, everyone makes a mistake sometimes.
The noun 'der Fehler' (the mistake/error) is closely related to the verb 'fehlen' (to be missing). A mistake describes a place where something is missing in order for it to be correct or complete.
In deinem Aufsatz fehlt ein wichtiges Argument.
An important argument is missing in your essay.
The context makes clear what kind of 'Fehler' (mistake/error) is meant. If you are talking about an action, it is a 'Fehler' (mistake). With a device, it is a 'Fehler' (defect). With a person, it can be a character weakness (flaw).
Einen Fehler in der Berechnung zu finden, ist manchmal schwer.
Finding a mistake in the calculation is sometimes difficult.
When referring to a wrong action, 'Fehler' is often combined with verbs such as 'zugeben' (to admit), 'einsehen' (to recognize/realize), 'korrigieren' (to correct), or 'bereuen' (to regret).
Es ist wichtig, seine eigenen Fehler einsehen und korrigieren zu können.
It is important to be able to recognize and correct your own mistakes.
When a defect is meant, 'Fehler' is often described more specifically with adjectives such as 'technisch' (technical), 'mechanisch' (mechanical), 'gravierend' (serious), or 'klein' (small).
Wegen eines kleinen technischen Fehlers musste das Flugzeug umkehren.
Because of a small technical error, the plane had to turn back.
problem
When an application is not working properly, both Problem and Fehler can refer to the difficulty being reported. Problem describes the disadvantageous situation in general, while Fehler additionally suggests that something deviates from the correct behavior. Use Problem when the consequence is the focus, and Fehler when the specific deviation is meant.

Das Programm hat einen Fehler und berechnet die Summe falsch.
The program has an error and calculates the sum incorrectly.
Das Programm hat ein Problem und startet heute nicht.
The program has a problem and will not start today.