Both Titania and Oberon are arguing over who gets to claim ownership of the Indian boy. Titania knew the boy's mother, who was a faithful servant to her, perhaps even a priestess (she describes the woman as being "of my order"). She describes the two of them as having spent time together gossiping in the evenings. However, the woman died in childbirth, so Titania has opted to bring up the boy for her servant's sake.
Oberon wants the child for a "knight of his train," as Puck puts it, though there isn't much in the text that suggests he is interested in the boy for his own merits. Oberon has no sentimental claim over the child as Titania does and seems to want him out of jealousy for his wife alone. He thinks Titania is giving the young child too much attention (Puck describes her as lavishing the boy with flowers and affection—much as she does with Bottom later on in the story, in fact) and therefore ignoring him. Some have argued that Titania's attentions have an erotic edge (once again, note the similarity of her behavior towards Bottom after she is enchanted with the love potion), thus prompting her husband's envy, but regardless, the point is she is no longer paying much attention to Oberon. Oberon also feels being Titania's husband and lord she should give him the boy just to please him. Naturally, Titania refuses out of loyalty to her dead friend.
This quarrel causes great consequences within the story. It's what inspires Oberon to use the love potion on his wife to make her fall in love with the transformed Bottom and then kicks off the disordered love square between the four young Athenians. By the end of the play, the effects of the love potion are reversed and Titania gives the child up to her husband.
Oberon and Titania, king and queen of the fairies, both want a young Indian boy to be part of their entourage. Titania wants the child, who is half mortal, half fairy, because she promised his dead mother, her friend, that she would raise and care for the child. Oberon wants the child so that Titania won't lavish so much attention on it. As Puck puts it:
And jealous Oberon would have the childKnight of his train, to trace the forests wild.But she perforce withholds the lovèd boy,Crowns him with flowers, and makes him all her joy
Matters come to an impasse. Oberon thinks Titania should obey him because he is her husband. She believes she should honor her promise to her friend. Their quarrel affects the weather:
Contagious fogs, which falling in the landHave every pelting river made so proudThat they have overborne their continents.The ox hath therefore stretched his yoke in vain,The ploughman lost his sweat, and the green cornHath rotted ere his youth attained a beard.
To gain the child, Oberon is willing to play a trick on a Titania, putting a love potion in her eyes so that she falls in love with the first creature she sees, which happens to be Bottom. Bottom is also enchanted, so that he has an ass's head. Titania eventually capitulates and gives up the boy, so that peace is restored.
In Act 2, sc. 1, Titania and Oberon quarrel. Oberon is upset that Titania is spending so much of her time and attention on a changeling child she has brought with her from India. The boy's mother was one of Titania's ladies-in-waiting as well as one of her friends. The woman, who was mortal, died giving birth to the child, who is part fairy, and Titania, for the sake of their friendship, has taken the child ro raise. Titania feels an obligation to this boy and apparently dotes on him which has caused Oberon to resent the child. Oberon wants to bring the boy into his fold to be his "henchman", but Titania does not want that. She is not ready to give up the boy yet. Oberon figures that if Titania is consumed with love for someone - as the juice from the flower would cause - then he'll be able to take the boy from her because her attention won't be on the boy. Oberon doesn't care with whom or with what Titania falls in love; he just wants her attention diverted so he can take the boy. Then Titania will pay attention to him again.
Oberon steals the boy from Titania, and she wants him back due the very circumstances very well written about in the above answer.
The boy is the offspring of one of Titania's handmaidens that she was very fond of. The handmaiden died because she was mortal and Titania is rearing the boy for the sake of this handmaiden. Oberon is jealous of the boy because he thinks the boy is receiving too much of Titania's attention and that he, Oberon, isn't receiving enough of it.
They both want the boy for very different reasons, part of the reason for the fight. Titania has more noble reasons in mind; she believes she owes it to her friend to raise the child "correctly" as his own mother would have. Oberon would like to have the boy as another personal jester, much like the job Puck holds throughout the play.
In act Two scene one, Oberon and Titania are fighting about Titania’s refusing to relinquish the changeling (a child secretly exchanged for another in infancy). She brought the child from India. His mother had been her friend and died in childbirth. Each accuses the other of infidelities and each takes a turn at denying these accusations.