die Bemühung
effort
An exertion or serious attempt to achieve a goal or complete a task.
Trotz grosser Bemühungen konnte die Mannschaft das Spiel nicht gewinnen.
Despite great efforts, the team could not win the game.

effort
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effort
An exertion or serious attempt to achieve a goal or complete a task.
Trotz grosser Bemühungen konnte die Mannschaft das Spiel nicht gewinnen.
Despite great efforts, the team could not win the game.

'Die Bemühung' (effort) is the noun related to the reflexive verb 'sich bemühen' (to make an effort). If you make an effort to do something, the result is your 'Bemühung' (effort).
Er bemüht sich, sein Deutsch zu verbessern.
He is making an effort to improve his German.
To describe what the effort refers to, an infinitive construction with 'zu' often follows. The structure is: 'die Bemühung, etwas zu tun' (the effort to do something).
Ihre Bemühung, einen Kompromiss zu finden, war deutlich zu sehen.
Her effort to find a compromise was clearly visible.
Common fixed phrases are 'trotz aller Bemühungen' (despite all efforts), when something did not work out, or 'vergebliche Bemühungen' (futile efforts), which means 'pointless efforts.'
Trotz aller Bemühungen konnten sie das Feuer nicht unter Kontrolle bringen.
Despite all efforts, they could not bring the fire under control.
The plural form 'Ihre Bemühungen' (your efforts) is a very polite and formal way to thank someone for their help or commitment. You often find it in business letters or emails.
Sehr geehrte Frau Meier, vielen Dank für Ihre Bemühungen in dieser Angelegenheit.
Dear Ms. Meier, thank you very much for your efforts in this matter.
operation / effort / use
'der Einsatz' and 'die Bemühung' can both describe someone making an effort to achieve a goal. 'der Einsatz' often sounds stronger and more positive: you give a lot of energy, time, or strength. 'die Bemühung' emphasizes the attempt and the effort more, even if the result is not certain. It is also often used in the plural, as in 'vielen Dank für Ihre Bemühungen' (thank you very much for your efforts), when someone has gone to trouble for another person. They can be confused because both mean effort. 'Einsatz' fits especially well with active, strong commitment; 'Bemühung' fits better with a serious attempt or polite thanks.

Herr Wagner dankt der Ärztin für ihre Bemühungen vor dem Termin.
Mr. Wagner thanks the doctor for her efforts before the appointment.
Ohne den Einsatz der Nachbarn wäre das Sommerfest nicht möglich gewesen.
Without the neighbors' effort, the summer festival would not have been possible.