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.

'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.
completion / degree
'der Abschluss' and 'die Erledigung' can both mean that something is brought to an end. 'der Abschluss' is more general and fits processes, events, or longer phases. 'die Erledigung' is used especially for tasks, work, or duties that you do and then check off. Learners may confuse the words because both sound like an ending. In practice: for a task, you often say 'Erledigung'; for a project, course, or conversation, 'Abschluss' is more likely.

Für die Erledigung der E-Mails braucht sie heute fast den ganzen Vormittag.
She needs almost the whole morning today to deal with the emails.
Nach dem Abschluss des Projekts feierte das Team gemeinsam.
After the project was completed, the team celebrated together.