Race is an important element in "Othello." It is one of the factors that Othello feels makes him an outsider, someone who is on the periphery of social and political power. The doubt that he experiences is, in part, due to the fact that he is Black. Iago works on this in suggesting that him being of color will impact his interactions with others, including Desdemona. The fact that Othello is Black is significant because he feels this is part of the reason why he could be undermined, and contributes to his own lack of confidence. I think that the issue of race is a powerful one as Shakespeare understood that color and ethnicities can be used as social constructs that can be made to play havoc with one's own sense of self. This is not to say that race is the reason why Othello's narrative is shaped the way it is, but rather to assert that race and the implications of being an individual in the position of power who is of color helps to feed Othello's own self- destructive nature rooted in fear, doubt, and insecurity.
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. akannan rightly points out above that "color and ethnicities can be used as social constructs that can be made to play havoc with one's own sense of self." I would add that it is worthy of examination how other characters in the play use the concept of race and fixed notions of what they take this to mean to play havoc with the sense of self of Othello - that is to say that race is a contested term that is used against Othello by others as well as by himself.
The play Othello was written in 1600, around the same time as the beginning of the African slave trade in Europe. In England, there were great fears of the Moors' Muslim threat to Christendom and the black man's quest for the symbol of white purity, the white man's woman.
Othello represents both of these fears within the play and upon the audience watching. As a black man who marries a white woman, the white public must suspect, like Brabantio, that she was seduced by sexual or magical powers.
The play produces two great taboos never seen on stage before: a black man married to a white woman and a bedroom murder scene. Both the marriage and the murder were scandalous to audiences then, and emotions must have raged when Othello strangled Desdemona in the last act.
White audiences in England were fascinated with tales from Africa, as evidenced in Leo Africanus' voyages and tales. He writes:
The Negroes likewise leade a beastly kinde of life, being vtterly destitute of the vse of reason, of desteritie of wit, and of all artes. yea they so behaue themselues, as if they had continually liued in a forrest among wilde beasts. They haue great swarmes of harlots among them; whereupon a man may easily coniecture their manner of liuing; except their conuersation perhaps be somewhat more tolerable, who dwell in the principall townes and cities: for it is like that they are somewhat more addicted to ciuilitie.
Like Iago and Brabantio, Leo paints them as jealous savages who treat women cruelly. Othello, during the play, seems to internalize these fears himself. In Act I, he nobly defends himself against racist fears in court. He uses language to defeat fear.
Later, however, Othello's jealousy rages when Iago exposes his racial sense of inferiority. In the end, it is a tragic self-fulfilling prophecy: Othello becomes the savage that the public has been trained to fear.
One of the first instances where race becomes important, and where it actually sets the theme for the rest of the novel, is when Rodrigo and Iago call out to Brabantio to inform him that his "white ewe" is being tupped by a "black ram." The idea is that Othello is a moor, and even though he is beloved by the people and the rulers of Venice, he is an outsider because of his skin color, and this is a division that Othello feels as well.
Othello worries that Desdemona's love for him might be inconstant since he is black and she might go for Cassio because he is white. It is one of the things that Iago manipulates to get Othello to go nuts.