erledigen
to take care of
To bring a task, job, or duty to an end or complete it.
Ich muss heute noch einige wichtige E-Mails erledigen.
I still have to take care of some important emails today.

to take care of / to finish
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to take care of
To bring a task, job, or duty to an end or complete it.
Ich muss heute noch einige wichtige E-Mails erledigen.
I still have to take care of some important emails today.

The prefix 'er-' is inseparable. This means it remains firmly attached to the verb in all sentence forms and is not placed at the end of the sentence.
Sie erledigt ihre Hausaufgaben immer sofort.
She always takes care of her homework right away.
The verb consists of the prefix 'er-', which often indicates a result, and the stem 'ledig', which means 'free'. So 'erledigen' (to take care of/complete) literally means to make something 'free' or 'finished'.
Sobald ich diese Aufgabe erledigt habe, habe ich frei.
As soon as I have taken care of this task, I am free.
In everyday life, 'erledigen' (to take care of/complete) is often combined with tasks and duties. Typical phrases are 'Arbeit erledigen' (to get work done), 'Aufgaben erledigen' (to complete tasks), 'Einkäufe erledigen' (to do shopping), or 'Besorgungen erledigen' (to run errands).
Am Samstag muss ich noch ein paar Besorgungen in der Stadt erledigen.
On Saturday, I still have to run a few errands in town.
The reflexive form 'sich erledigen' (to take care of itself) means that a problem disappears on its own. The subject of the sentence is always the matter or problem, not a person.
Das Problem mit dem Drucker hat sich von selbst erledigt.
The problem with the printer took care of itself.
In the form 'erledigt sein' (to be wiped out), the past participle is used like an adjective and means 'completely exhausted'. This use is very common in colloquial language.
Nach dem langen Arbeitstag bin ich total erledigt.
After the long workday, I am totally wiped out.
to do / make
'machen' and 'erledigen' can both be used for tasks or work. 'machen' only says that you carry out an activity. 'erledigen' emphasizes that a task is completed and is no longer unfinished afterward. That is why you can say 'Ich mache die Hausaufgaben' (I am doing the homework), while 'Ich erledige die Hausaufgaben' more strongly means that you get it finished. Learners can easily confuse the words because both fit with tasks. If finishing the task is important, 'erledigen' is more precise.

Bitte erledige die Anmeldung noch vor dem Termin.
Please take care of the registration before the appointment.
Das Kind macht nach der Schule seine Hausaufgaben.
The child does his homework after school.