I would have to go with the theme of duality as one of Shakespeare's most important ideas in the tragedies: what appears to be reality is often not. Notice how often the word "seem" is used in the play: "Men should be what they seem." This familiar lament is ironically stated and discussed repeatedly by Othello and with, ironically, the most two-faced character in the play. It happens over and over in the plays. Here, it ends in total disaster for Othello, who is far too trusting to understand the reality of Iago. Othello's simplistic military code is no match for Iago's Machiavellian manipulations.
The second most important theme would then be the nature of jealousy, one of our most primitive emotions. Shakespeare explores its permutations with 3 different couples, all of whom have one partner who is fairly consumed by it and the other who deals with it as it affects their relationship. Ultimately, it destroys or at least seriously hurts all 3 couples. Desdemona and Emilia are basically killed because of it. Shakespeare has many penetrating observations about this emotion. The one that strikes me is that he says that jealous souls are never jealous for a reason, but because they are jealous. In other words, there is no good reason for them to be jealous. They just are. That is scary stuff and well worth considering as a warning.
The difficulty in controlling doubt and the intensely complex landscape of human emotions strikes me as thematic fields of play in "Othello." The exploitation of Othello's insecurity and the manipulation engineered by Iago are the most powerful levels where themes are apparent for me. Iago's own jealous nature and his resentment are present and the theme I get from this is how individuals can be motivated by their own subterranean demons. Iago sets out to destroy everyone's own sense of happiness only because he was denied his. For his part, Othello, despite his accomplishments and endeavors, is completely trapped by his own demons of insecurity and doubt, making him completely pliable for Iago. The theme of how human beings must master doubt and insecurity by living in peace with them as opposed to overcoming them might be the most relevant theme in such a dynamic.
Though race is often brought up in this play, the more accurate differentiation is class. Cassio has received the position of lieutenant instead of Iago because of Cassio's social standing. Othello was obtained by Florence as a slave, but was the son of a chieftain. As he feels right in having Desdamona as his wife, he is in tune with social order and the superiority of those of noble birth.
Despite their commonality and mutual respect, the Duke and Brabantio always refer to Othello as "the Moor," thus keeping him separated from the rest of their society. This, ironically, turns on Othello as he has passed up Iago for the position. Iago is certainly the more cunning of the two choices, Cassio being much more of a pretty boy of good breeding.
I would say the most important themes are
-truth and false
-how the women were treated
-the superiority of men
-balck and white