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 child
Knight 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 land
Have every pelting river made so proud
That they have overborne their continents.
The ox hath therefore stretched his yoke in vain,
The ploughman lost his sweat, and the green corn
Hath 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 Shakespeare's A Midsummer Night's Dream, why are Oberon and Titania fighting?

Puck sets forth a few of the details of Oberon's and Titania's fight to an unnamed fairy in Act II Scene i:

"For Oberon is passing fell and wrath,/ Because that, she as her attendant, hath/ A lovely boy stolen from an Indian king./She never had so sweet a changeling./ And jealous Oberon would have the child/ Knight of his train, to traces the forests wild./ But she perforce withholds the loved boy,/ Crowns him with flowers, and makes him all her joy" (II.i. 20-27).

Basically, Titania has a young boy who is getting all of her attention like a mother would give and Oberon is jealous. In addition, Oberon wants the boy to be his so he can train him like a soldier in the wild. Later, Oberon tells Titania that all he wants is for the boy to be a henchman (servant) for him. Titania explains that she has claim to the boy because his mother worshipped her but died at child birth. From that point on, Titania has protected and raised the boy in his mother's loving memory. Here we see that Titania loves the boy like a mother, has raised him since he was a baby, and Oberon selfishly wants to take the boy away from her. To Oberon, the boy is not his, so it would be understandable that he wouldn't value the boy like a son. Since Oberon is not the boy's father, however, he doesn't have much of a say as to what should be done with him. As the custody battle ensues, Oberon decides to drug Titania and steal the boy that way.

In A Midsummer Night's Dream, why is Oberon unhappy about Titania adopting the Indian boy?

The changeling, or Indian boy that Queen Titania stole from an Indian king and replaced with another child, was a particularly sweet and beautiful child, as we see from Puck's line, "She never had so sweet a changeling" (II.i.23). Since the boy is so beautiful, Oberon is actually angry with the queen for adopting him because Oberon is jealous. Oberon wanted the boy for his own care because he wanted to train him to be one of his knights, as Puck explains in his line "[a]nd jealous Oberon would have the child / knight of his train, to trace the forests wild" (24-25). Oberon has asked the queen for the boy, but she refuses, which has made Oberon extremely angry, as Puck explains in the lines,"But she perforce withholds the loved boy / Crowns him with flowers, and makes him all her joy" (26-27). Oberon and the queen keep having such violent fights about the issue that the elves hide in acorns out of fear, as we see in the line, "[A]ll their elves for fear / Creep into acorn cups and hide them there" (30-31).

Hence, we see that the reason Oberon is angry over the adoption is that he is jealous and wants the boy for himself.