A rich and powerful man, Brabantio isn't used to people going against his wishes, especially not his own daughter. In common with the prevailing social standards, Brabantio enjoys complete control over his family; he is a forbidding patriarch whose word has the status of holy writ.
So imagine his utter shock when he finds out that his daughter Desdemona has eloped with Othello and married him. Brabantio can't quite believe his own ears. He's so shocked that he actually thinks that Desdemona must somehow have been drugged or put under an evil magic spell.
Brabantio's racism is much in evidence here. He's attributing all kinds of dangerous, exotic powers to Othello, simply because he happens to be black. The very idea that his own flesh and blood chose to run off with Othello of her own free will simply doesn't occur to him; the very thought of it is too horrible to contemplate.
Brabantio feels like he's been deceived by Desdemona, and he's certain that Othello will eventually suffer the same fate:
Look to her, Moor, if thou hast eyes to see.
She has deceived her father, and may thee. (1.3.333–334)
Of course, Desdemona won't betray Othello, but Othello will come to think that she has, with tragic consequences. But not before a traumatized Brabantio dies of grief at what he sees as his daughter's betrayal.
When Roderigo and Iago warn Brabantio that Desdemona has married Othello, he's upset and doesn't believe it's true.
Roderigo and Iago tell Brabantio that his daughter is sleeping with Othello in order to upset Brabantio and further damage Othello out of jealousy and racism. Her father doesn't believe it at first but then wakes his household so that he can investigate. He realizes she's gone, says that she must have been tricked or charmed into it, and leaves to make Othello accountable. (This takes place in Act 1, Scene 1 of the play.)
In the next scene, Brabantio confronts Othello. He accuses him of theft and witchcraft. Brabantio asks the guard to arrest Othello. The men argue but find out that the Duke has called them both to a meeting.
In Act 1, Scene 3, the men attend the meeting with the Duke. Brabantio charges Othello with witchcraft; Othello explains that Desdemona fell in love as he listened to his tales of his bravery. As Brabantio learns more about Othello, his temper fades somewhat. He isn't satisfied with the marriage, but he is more accepting. He does begrudgingly give his blessing.
Brabantio is certainly displeased at hearing the news that Desdemona and Othello plan to be married. Brabantio reacts in a typically racist, xenophobic, and patriarchal manner in which he is incredibly upset that Desdemona would wish to marry outside her race/nationality and, additionally, feels that she has been stolen from him, as if Desdemona is some kind of object that can be taken away from its rightful owner. Brabantio feels that the marriage goes against what he deems to be "natural" and is convinced that Desdemona must have been coerced by drugs or sorcery into agreeing to the marriage. When he finally speaks to her, however, he realizes that she does, in fact, love Othello, and that he can not stop the union from happening.
Brabantio has an angry, unhappy reaction to his daughter Desdemona marrying Othello. When he hears that Desdemona has eloped with Othello, he states it is unnatural and shows poor judgment to marry a person different from her in so many ways, such as "years" (age), nationality ("country) and race ("what she feared to look on"):
She, in spite of nature,
Of years, of country, credit, every thing,
To fall in love with what she feared to look on!
It is a judgment maimed and most imperfect
Brabantio also states:
She is abused, stol'n from me, and corrupted
He comes to the Duke in a state of fury, wanting Othello punished or even killed for stealing his daughter. He believes that Othello used some sort of cunning witchcraft or sorcery or perhaps drugged Desdemona in order to get her to marry him.
Brabantio, however, calms down as he learns that Desdemona fell in love with Othello as she listened to the stories he told her father of his warrior exploits. He accepts the marriage when he questions her, and she says this is what she wants. Nevertheless, he is still not happy. He no longer accuses Othello of having stolen her but, he says:
I am glad at soul I have no other child.
Brabantio is extremely displeased at the idea that his daughter has married the Moor. He had a good relationship with Othello and often spent time with him and loved to hear the stories of his campaigns and the wars in distant places. Brabantio was very happy with Othello in the role of general of Venice's armies, but he was absolutely not prepared to have his daughter marry the man.
He goes so far as to suggest that Othello must have used some kind of witchcraft to convince his daughter to marry him. He feels that it is unnatural for a Venetian woman (white) to marry the Moor (black) and makes it very clear that he does not approve. He appeals to the Duke for justice in the matter, suggesting again that Othello used witchcraft to win his daughter's hand.
Only after Othello and Desdemona explain the story of their courtship does Brabantio relent, and even then it is with hesitation. He even warns Othello that Desdemona is not to be trusted, since she tricked him, she might also deceive Othello.