What is the significance of Othello's race in Othello by William Shakespeare?

Race is central to the plot of Othello in several ways, both some that make the play distinctly of its period and some that make the play appear modern. It is important for readers to keep both in mind simultaneously, rather trying to read the play as a modern civil rights or post-colonial narrative.
Othello is a Moor. What that means is somewhat ambiguous. One common use of the term would have been to refer to Arabs in North Africa and Spain. It might also apply to people from sub-Saharan Africa, but that usage is somewhat less common. Moorish people in the sense of Arabs were well known in Europe, with a Moorish community actually present in London in Shakespeare's lifetime, and thus the likely referent.
Most literature in Shakespeare's period accepted the notion that different ethnic groups had different types of character, with Northern (white) Europeans being rational, self-controlled, and highly civilized but people (with darker skins) from warmer climates being more passionate by nature. Thus race is significant in that the original audience would have been (by twenty-first century standards) racist and assumed that under Othello's civilized and admirable character lurked his racial character of intense pride, jealousy, passion, and a tendency to be emotionally volatile rather than rational.
The next aspect of race in the play is the racism that compounds Iago's jealousy and makes Brabantio prejudiced against Othello as a possible husband for Desdemona, despite Othello's career success and the couple's love for each other. Othello encounters racial prejudice in several contexts in the play, overcoming prejudice through his military skill and noble character.

Othello's status as a black man in a white man's world is of vital importance to the plot of Shakespeare's Othello. In defiance of stereotypical and racist assumptions about the black community during Shakespeare's day, Othello is a dignified, honorable man who enjoys considerable power and influence. Moreover, Othello proves himself to be an eloquent and gifted speaker, as he consistently employs sophisticated rhetoric when communicating with his peers. Clearly, Othello deconstructs the racist stereotypes that characterize black individuals as dangerous or uncivilized beings. The fact that he does so invokes the hatred of those white individuals unwilling to accept a black man with intelligence, power, nobility, and dignity. For example, though Iago seeks revenge because he wants the promotion Othello gives to Cassio, we can assume there's a racist subtext underlying this primary motive. Indeed, it's possible to see Iago's plot as the racist effort to oppress a black man who has attained power in a white society. As such, we can see that Othello's race is vitally significant to the play's plot.