die Exekution
execution
The killing of a person as the carrying out of a death sentence.
Die Exekution des Verurteilten fand im Morgengrauen statt.
The execution of the convicted man took place at dawn.

execution
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execution
The killing of a person as the carrying out of a death sentence.
Die Exekution des Verurteilten fand im Morgengrauen statt.
The execution of the convicted man took place at dawn.

The noun 'Exekution' (execution) is derived from the verb 'exekutieren' (to execute), which means 'ausführen' (to carry out) or 'vollstrecken' (to enforce/carry out a sentence). The ending '-tion' is typical of nouns that describe a process or action and come from Latin.
Der Befehl wurde sofort exekutiert.
The command was executed immediately.
You can recognize the meaning of 'Exekution' (execution) from the context. In connection with a 'Urteil' (judgment/sentence) or 'Verurteilten' (convicted person), it means an execution. With 'Gericht' (court) and 'Schuldner' (debtor), it refers to enforcement of a judgment. In the field of 'Computer' or 'Code', it means program execution.
Die Exekution des Befehls dauerte nur wenige Millisekunden.
The execution of the command took only a few milliseconds.
In the meaning 'Hinrichtung' (execution), 'Exekution' (execution) is a very formal and official term. It is used especially in historical, legal, or journalistic texts.
Historische Dokumente beschreiben die öffentliche Exekution im Mittelalter.
Historical documents describe the public execution in the Middle Ages.
This meaning is a technical legal term. If you are talking about the forced collection of debts, 'Zwangsvollstreckung' (enforcement of a judgment) is the more common word in everyday language.
Gegen die Firma wurde die Exekution eingeleitet.
Enforcement proceedings were initiated against the company.
In computer science, 'Exekution' (execution) is a technical term for running code. In general everyday language about computers, people usually simply speak of the 'Ausführung' (running/execution) of a program.
Während der Exekution des Programms trat ein unerwarteter Fehler auf.
An unexpected error occurred during the execution of the program.
implementation / feedthrough
'Durchführung' and 'Exekution' can both have to do with carrying something out or implementing it. 'Durchführung' is the normal, neutral word for plans, measures, or events. 'Exekution' is much more specific. In computer science, it can mean the execution of a program; in some specialist fields, it can mean technical execution; and in law, it can mean the compulsory enforcement of a claim. It also has the very serious meaning of execution in the sense of putting someone to death. For that reason, you should not simply use 'Exekution' as a replacement for 'Durchführung'. The risk of confusion arises because both words can sometimes mean 'Ausführung' (execution / carrying out), but in everyday language 'Durchführung' is usually safer and more natural.

Bei der Exekution des Programms zeigt der Computer eine Fehlermeldung.
During the execution of the program, the computer displays an error message.
Die Durchführung des Sprachkurses hat gut funktioniert.
The language course was carried out successfully.