In Act I of Othello, Iago is disloyal to Othello as he exposes his secret marriage to Desdemona by summoning Brabantio. Brabantio is disloyal to Othello because he had liked him up until he married his daughter. The Duke is the only one loyal to Othello, as he makes him General to Cyprus and even lets Desdemona accompany him. Desdemona is disloyal to her father, as she deceives him by eloping. Iago is disloyal to Roderigo, as he openly says "I am not what I am" and just uses him for money. Othello is loyal to all, although Iago complains that he was passed over by the Moor to be his lieutenant. Disloyalty to women and loyalty to male reputation, to me, are the primary tragic mistakes in the play.
In Acts II-IV, Iago ensnares two prime victims: Cassio and Desdemona who only APPEAR to be disloyal to Othello. Othello puts too much faith in the possession of the handkerchief, and nearly everyone has their hands one it: Emilia, Iago, Cassio, and--worst--Biancha, a prostitute. When Othello sees this, he effectively divorces Desdemona and swears a blood oath to Iago to commit double murder against the adulterers.
In Act V, as the previous editor said, Desdemona is loyal to Othello to a fault. Her foil is Emilia, who is disloyal to Iago, which, of course, makes her loyal to the memory of Desdemona. Both are killed by their husbands. Loyal or disloyal to husbands, either way, women are doomed in this play.
Shakespeare would have us believe, then, that his society was riddled with sexism, misogyny, and double standards in marriage that state men can talk openly and have affairs openly, but women can do neither.
One of the driving forces behind the actions of Iago, Othello, as well as even Cassio and Desdemona revolve around their loyalties and the betrayals, mainly those perpetrated by Iago. The loyalty to Iago is what leads Othello to his tragic destruction and Iago uses that to his utmost advantage.
Desdemona is loyal to Othello to a fault, and has difficulty challenging him or expressing her worries about him to anyone besides her nurse. Her nurse's loyalty is what drives her to be the one that will discover much of the action and expose some of the treachery before her death at the hands of Iago.
Betrayal, by Iago, of everyone and everything that he is ostensibly loyal to, Othello, Cassio, the army, his friends, etc., is even brought about by what he says as disloyalty to him despite all his loyal service when Othello chooses Cassio over him as his new Lieutenant.