Othello does not deal with issues of prejudice and discrimination--that's the ironic part of the whole play. One might think that a character like Othello, often (at least behind his back), the brunt of discriminatory comments--such as "thick lips" and the innumerable references to him and his sexuality as being bestial), would be more sensitized to prejudice, but he is not. Iago suggests that the women of his country are all of ill-repute, and Othello falls right into Iago's trap and thinks that all Venetian women, Desdemona included, are strumpets (prostitutes) at heart, as the stereotype suggests. He does not think that Desdemona could be anything but a subtle "whore of Venice", even though, by his own admission, she looks nothing like a prostitute. He buys into the stereotype so well that he calls Emilia simple when she says that Desdemona is virtuous. Instead of questioning his assumptions, he says he thinks that Desdemona must be a very cunning whore to trick Emilia as well as him. Prejudice and discrimimation are rampant in the play, not only from a black and white perspective, but also from the perspective of class and gender.
Othello's race and Shakespeare's response to it goes beyond color; Shakespeare responds to Othello's "otherness." He comes from a mysterious "other world." Even though he defines himself as Venetian and Christian, he cannot shake off this exoticism. From an Elizabethan viewpoint, Othello's rank as governor of Cyprus would intensify his otherness. It is as if Caliban, dress to the nines, were to reappear as Prospero, king of the island. That a man from such an exotic place (absolutely foreign) lords it over Europeans (when he talks about his background) let alone marries an upper-class white wife, upsets all contemporary notions of decorum. Iago makes various racist remarks about the color of Othello's skin, to be sure, but it is his "otherness"--being from a foreign, mysterious, exotic place--that Shakespeare pays most attention to and sees as part of the problem for his reluctant inclusion in dominant, nice society.
Shakespeare writes about these themes as they were present in his own England while setting his play in Venice (where it was probably present as well). As with his treatment of women, Shakespeare shows himself to be a man beyond his years in his approach to issues like prejudice and discrimination. He condemns these evils rather than glorifying them, which was truly amazing for his time.
The link below to modern connections to Othello will be very helpful in answering this question, as will the links to information on the characters of Brabantio and Iago. Brabantio (Desdemona's father) and Iago are the characters Shakespeare uses to highlight the ignorance and barbarity of prejudice and discrimination in this play. Good luck!