Race provides the most prominent catalyst for the conflicts that become evident in the play. If not for Othello's race, he might be immune, or at least more resistant, to Iago's manipulation. Othello is keenly aware that his skin color creates a barrier that he can never completely overcome in his dealings with the Europeans. Othello will always be an outsider in Venice, no matter how heroic his actions or wise his counsel.
I agree that race is important in that it is significant to Othello himself. It is his achilles' heel and he cannot transcend the boundary he perceives between himself and others. Tragically his wife Desdemona was unable to persuade him to see beyond his colour as she does, but as others' have said elements of the society at the time, and Othello's own history conspire to make race a significant factor in Iago being able to lead him to his downfall.
Just to add to the above points, I think it is of interest that race is so strongly linked to the concept of identity....
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