"Faith" comes from the Anglo-French "feid," from Latin "fidēs," which means "trust" and "confidence." In Latin, "bona fide" means "in good faith," or honest intention.
"Faith" is mentioned 21 times in the play. Most of the time it is used in the phrase "by my faith," "in faith," "I' faith," and "good faith," which all mean "really."
Cassio, Iago, Othello, Desdmona, Montano, Emilia all use it. It's an adverb used as a desperate means to convince others and oneself of credibility, truthfulness. Look at the way Cassio uses it to convince Iago that the prostitute Biancha loves him:
Iago: Will you sup there?
Cassio: Faith, I intend so.
Speaking to the Duke, Othello says of Desdemona:
Faith also means faithfulness and fidelity in marriage. Othello's closing remarks to the Court again uses "faith" as a means of persuasion:
This admission seals Othello's fate. So says Annika Odland:
Does he mean his faith in her or her faith in him, or her Christian faith which in Venice also includes obeying her father? If the latter alternative is the case than his faith in her may not last very long. By deceiving her father she has proved she is not trustworthy.
Faith is a counterpoint to honest. Iago is called "honest" something like 27 times in the play. He calls himself honest; Cassio and Othello, his enemies, call him honest. In this society, men were, by default, considered to be honest; women were expected to be faithful. A classic double standard.
Othello wrests his reputation upon the fidelity of his wife. And although she remains faithful to him throughout the play, Othello cannot abide an unfaithful wife, such was the rigid power structure of the "culture of honor." Othello performs and honor killing to protect his image.
A lack of faith on all levels (religious, interpersonal, cultural, societal) dooms most characters in the play. Othello loses faith in himself, his wife, his race, his status, his ability to communicate, his friends, his ability to lead. Desdemona loses faith in her father, in her role as wife, in her position in society, in her gender. Cassio loses faith in his general, himself, his reputation, his treatment in women.