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.

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.
damage / harm
'der Fehler' and 'der Schaden' can both have to do with something negative. A 'Fehler' is something that was done incorrectly, or a defect in something. A 'Schaden' is the negative consequence, for example something broken, a loss, or a disadvantage. They are often confused because a 'Fehler' can cause a 'Schaden'. But they do not mean the same thing: the 'Fehler' is often the cause, while the 'Schaden' is often the result.

Im Bericht hat Lisa einen Fehler entdeckt und ihn sofort korrigiert.
Lisa found a mistake in the report and corrected it right away.
Nach dem Sturm war der Schaden am Dach deutlich zu sehen.
After the storm, the damage to the roof was clearly visible.