
absagen
to cancel
To cancel or declare invalid a scheduled appointment, event, or arrangement.
Wegen Krankheit musste sie das Konzert leider absagen.
Because of illness, she unfortunately had to cancel the concert.
to cancel
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to cancel
To cancel or declare invalid a scheduled appointment, event, or arrangement.
Wegen Krankheit musste sie das Konzert leider absagen.
Because of illness, she unfortunately had to cancel the concert.

to decline
To decline an invitation or an offer.
Er hat zur Party abgesagt, weil er schon etwas anderes vorhat.
He declined the party invitation because he already has something else planned.

to turn someone down
To give a person a negative answer to an application or request; to refuse acceptance or approval.
Das Unternehmen hat allen anderen Bewerbern abgesagt und ihr die Stelle gegeben.
The company turned down all the other applicants and gave her the job.
'Absagen' (to cancel/decline) is a separable verb. The prefix 'ab-' goes at the end of the main clause. In subordinate clauses and in the infinitive with 'zu', the verb stays together.
Ich sage den Termin für morgen ab.
I am canceling the appointment for tomorrow.
When you cancel something planned, 'absagen' (to cancel) needs an accusative object. Typical objects are: 'einen Termin' (an appointment), 'eine Reise' (a trip), 'eine Party' (a party), or 'ein Treffen' (a meeting).
Wir mussten unseren Flug wegen des Streiks absagen.
We had to cancel our flight because of the strike.
When declining an invitation, 'absagen' (to decline) can also be used without an object. The context then makes clear what it refers to.
Danke für die Einladung, aber ich muss für Samstag leider absagen.
Thanks for the invitation, but unfortunately I have to decline for Saturday.
When you give a person a negative answer (e.g., to an application), the person is put in the dative. The structure is: 'jemandem absagen' (to turn someone down).
Die Firma hat ihm leider für die Stelle abgesagt.
The company unfortunately turned him down for the position.
to reject
Ablehnen and absagen can both express a negative response to an invitation or an offer. Ablehnen tends to be directed at the suggestion itself. Absagen is used especially when an invitation is declined or an appointment that has already been arranged is canceled.

Mara sagt den bereits vereinbarten Termin wegen einer Krankheit ab.
Mara cancels the appointment they had already arranged because of an illness.
Mara lehnt den vorgeschlagenen Termin ab, weil er ihr nicht passt.
Mara rejects the proposed appointment because it does not work for her.