
die Überzeugung
conviction
A firm inner certainty or strong belief in the correctness of an idea, opinion, or fact.
Ich handle aus der tiefen Überzeugung, das Richtige zu tun.
I act out of the deep conviction that I am doing the right thing.
conviction / persuasion
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conviction
A firm inner certainty or strong belief in the correctness of an idea, opinion, or fact.
Ich handle aus der tiefen Überzeugung, das Richtige zu tun.
I act out of the deep conviction that I am doing the right thing.

to come to the conviction
The process of arriving at a firm opinion or certainty through reflection, arguments, or evidence.
Nachdem er alle Fakten geprüft hatte, gelangte er zur Überzeugung, dass der Plan funktionieren würde.
After he had checked all the facts, he came to the conviction that the plan would work.

persuasion
The process of getting another person, through arguments, to accept a particular opinion or belief.
Die Überzeugung des skeptischen Kunden erforderte viel Geduld und gute Argumente.
The persuasion of the skeptical customer required a lot of patience and good arguments.
'Die Überzeugung' (conviction/persuasion) is the noun related to the verb 'überzeugen' (to convince). It describes the result of convincing: the state of firmly believing in something.
Seine Argumente waren so gut, dass sie mich überzeugt haben.
His arguments were so good that they convinced me.
To describe the content of the conviction, a subordinate clause with 'dass' (that) or an infinitive construction with 'zu' (to) is very often used.
Ich bin der festen Überzeugung, dass wir dieses Projekt schaffen werden.
I am firmly convinced that we will complete this project.
Common adjectives that emphasize the strength of the belief are 'feste Überzeugung' (firm conviction) or 'tiefe Überzeugung' (deep conviction). Another typical phrase is 'aus Überzeugung handeln' (to act out of conviction).
Sie ist Vegetarierin aus tiefster Überzeugung.
She is a vegetarian out of deepest conviction.
The process of forming an opinion is often expressed with the fixed phrase 'zur Überzeugung gelangen' or 'zur Überzeugung kommen' (to come to the conviction).
Nach langem Nachdenken kam ich zur Überzeugung, dass ein Umzug die beste Lösung ist.
After thinking about it for a long time, I came to the conviction that moving is the best solution.
opinion
Both die Meinung and die Überzeugung refer to a personal view. A Meinung can still change, while a Überzeugung expresses a much firmer and more deeply rooted certainty. Use die Meinung for an ordinary position and die Überzeugung when someone is very certain.

Sie hat die feste Überzeugung, dass weitere Informationen ihre Entscheidung nicht ändern werden.
She firmly believes that more information will not change her decision.
Meine Meinung zu diesem Thema kann sich nach weiteren Informationen noch ändern.
My opinion on this topic may still change after I receive more information.