What do Othello and Desdemona find attractive about one another?
I think that the wording of the question is interesting. There is not a mention of "love." I find that fascinating because it is of considerable debate as to whether or not both of them loved one another. I think that Desdemona is attracted to the "outsider" status of Othello. This probably enters into as many categories as possible. He is accomplished as a warrior and military strategist, has seen the world over, is extremely powerful and with a physique to match, and is an outsider by Venetian social mores. He is a Moor and is Black. Certainly, I think that all of these provide an attraction for Desdemona. If it is true that she had considerable suitors who vied for her fancy, it can be presumed that most of them followed within the norms of Venetian society. Being of good social standing, it is likely that the men who would have competed for her wares were probably of typical social standing and all represented much of the same ilk. Othello was different and this probably captivated Desdemona. There is a reason that she leaves with Othello to Cyprus, and is about as far removed as possible from Venetian society. For his part, Othello is probably attracted to the fact that someone like Desdemona is attracted to him. This might be the idea that the ultimate outsider could not conceive of being found attractive by someone who is, for all practical purposes, the ultimate insider. Desdemona's whiteness and Othello's darkness is literal in this understanding. There is an attraction being found attractive, and this might be what holds Othello. It makes sense because part of what Iago operates upon is the idea that there is nothing that would be attractive in Othello to someone like Desdemona, making her deception all the more plausible. This does work on Othello as he doesn't doubt her as much as he doubts himself, as in why would someone like her like him.
Othello in his address to the duke explains how the two fell in love. He sums up his story with these lines:
She loved me for the dangers I had passed,
And I loved her that she did pity them.
This only is the witchcraft I have used.
When Othello visited Brabantio's house, he would tell stories of his adventures. Desdemona would listen with such interest and sympathy that the two fell in love. Desdemona, most likely, was quite sheltered, and was immediately attracted to the man who committed such heroic deeds. In turn, Othello delighted in Desdemona's interest and compassion for the dangerous life he had led. Shakespeare makes it clear through this speech that Othello and Desdemona's love was not an infatuation, that it grew over time, and it was based not on looks or superficialities but rather on personalities and history. Desdemona is presented as a mature, sensitive young lady while Othello is portrayed as nothing short of a hero.
Later Desdemona declares,
I saw Othello's visage in his mind,
And to his honors and his valiant parts
Did I my soul and fortunes consecrate.
Desdemona loves Othello not for his looks but for the way he thinks. Othello loves Desdemona because she exhibits the traits of the ideal woman of that time period: submissive, obedient, faithful, quietly spoken, accomplished in dancing, singing, and sewing. But Othello senses the basic goodness of Desdemona as he does initially with Cassio. Othello's instinctive judgments of those around him are probably sound: he did choose Cassio over Iago for his lieutenant. It is only when he begins to listen to Iago's insinuations that he begins to doubt himself and his own judgments of those he loves.