
die Erledigung
completion
The completion or carrying out of a task, a piece of work, or a duty.
Die Erledigung dieser Aufgabe hat den ganzen Vormittag gedauert.
The completion of this task took the whole morning.
completion / elimination
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completion
The completion or carrying out of a task, a piece of work, or a duty.
Die Erledigung dieser Aufgabe hat den ganzen Vormittag gedauert.
The completion of this task took the whole morning.

elimination
Colloquially and often euphemistically used for the killing or final defeat of a person.
Im Film übernahm der Auftragskiller die Erledigung des unliebsamen Zeugen.
In the film, the contract killer took over the elimination of the unwanted witness.
'Die Erledigung' (completion) is the noun corresponding to the verb 'erledigen' (to get something done, to complete a task). If you know the verb, you can easily work out the meaning of the noun.
Ich muss heute noch einige wichtige Dinge erledigen.
I still have to take care of some important things today.
In normal usage, 'Erledigung' (completion) usually refers to everyday or professional duties. Typical combinations are the completion of tasks, housework, shopping, or formalities.
Die schnelle Erledigung der Formalitäten hat mich überrascht.
The quick completion of the formalities surprised me.
To express what is being completed, German usually uses the genitive or the preposition 'von' with the dative. The genitive often sounds somewhat more formal.
Die Erledigung der Steuererklärung ist sehr komplex.
The completion of the tax return is very complex.
You can clearly recognize the meaning 'Tötung' (killing) from the context. It is used almost only in stories about crime or in action films and often sounds like a cold, impersonal order.
Der Boss befahl die sofortige Erledigung des Verräters.
The boss ordered the immediate elimination of the traitor.
implementation / feedthrough
Both Durchführung and Erledigung can refer to working on a task. Durchführung emphasizes the organized process or implementation, whereas Erledigung emphasizes successfully getting something finished. Use Durchführung for a project and Erledigung for an individual duty or task.

Die Erledigung der letzten Aufgabe dauerte nur zehn Minuten.
Completing the last task took only ten minutes.
Die Durchführung des Arbeitsplans dauerte mehrere Wochen.
Implementation of the work plan took several weeks.