effort / overcoming
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effort
The mental or physical effort needed to overcome an inner barrier, fear, or resistance. Often used in the expression 'etwas kostet Überwindung' (something takes effort / courage).
Es hat mich große Überwindung gekostet, vor so vielen Leuten zu singen.
It took a great deal of effort for me to sing in front of so many people.
'Die Überwindung' (overcoming / effort) is the noun related to the inseparable verb 'überwinden' (to overcome). The ending '-ung' turns the action of overcoming into a thing. The verb means to defeat a difficulty or obstacle.
Man muss seine Ängste überwinden, um frei zu sein.
You have to overcome your fears in order to be free.
This meaning is almost always used in the fixed expression 'es kostet jemanden Überwindung' (it takes someone effort / courage). The person who has to make the effort is in the accusative.
Es kostet mich jedes Mal Überwindung, vor Publikum zu sprechen.
It takes effort for me every time to speak in front of an audience.
In this meaning, what was overcome is often specified. This is usually done with a genitive, as in 'die Überwindung der Krise' (overcoming the crisis), or with 'von' + dative, as in 'die Überwindung von Hindernissen' (overcoming obstacles).
Die Überwindung seiner Spielsucht war ein großer Erfolg.
Overcoming his gambling addiction was a great success.
The expression 'es kostet Überwindung' always refers to an inner, psychological barrier. If, on the other hand, people speak of 'Überwindung von etwas' (overcoming something), e.g. an illness or an opponent, it means defeating an external problem.
Es kostete ihn Überwindung, von der hohen Brücke zu springen.
It took effort for him to jump from the high bridge.
victory
'der Sieg' and 'die Überwindung' can both describe managing to overcome a difficulty. 'der Sieg' emphasizes the successful result: you have won or left a problem behind you. 'die Überwindung' puts more emphasis on the path to get there, meaning the effort, fear, or inner barrier involved. For example, you can speak of a 'Sieg über die Angst' (victory over fear), but in everyday language people often say that something 'kostet Überwindung' (takes effort / courage). The words can be confused because both sound positive. 'Sieg' fits more with the result, while 'Überwindung' fits more with the inner effort.
Es kostet Emma viel Überwindung, vor der ganzen Klasse zu sprechen.
It takes a lot of courage for Emma to speak in front of the whole class.
Für Paul war die bestandene Prüfung ein Sieg über seine große Unsicherheit.
For Paul, passing the exam was a victory over his great insecurity.