What is the significance of gender in Othello?

Gender is extremely significant in Othello. Iago uses gender norms common in Venice at that time to manipulate Othello into killing the woman he loves.
The stage for this is set early in the play, when Iago hears Brabantio, Desdemona's father say to Othello in act I, scene 3:

Look to her, Moor, if thou hast eyes to see: / She has deceived her father, and may thee.

At this time, gender norms strictly dictated that a woman must be completely sexually faithful to her husband. But, at the same time, many men believed a prejudiced gender stereotype that said women were inherently unfaithful, always ready to have affairs with other men. This is implied in Brabantio's words, the words of a sober patriarch: women are untrustworthy. Iago himself seems to believe this as part of his generally dark view of human nature. He plays on it ruthlessly to plant doubts in Othello's mind about Desdemona.
Because of the honor code that said one's manhood was undermined if one's wife was unfaithful, and because it was considered acceptable to kill an unfaithful wife, Othello kills his innocent spouse. If gender rules had not been so strict, the play could have had a more reasonable outcome.