This is really two questions, but I'll try to answer them both.
Why don't I remember reading about Azog losing his arm?
Azog does not appear in the book, although he is mentioned by name in chapter one, An Unexpected Party:
Gandalf: Your grandfather Thrór was killed, you remember, in the Mines of Moria by Azog the Goblin.
Thorin: Curse his name, yes.
The incident Gandalf's referring to is called the War of the Dwarves and Orcs1, and it's described in Appendix A: Annals of the Kings and Rulers, part III "Durin's Folk".
The war is precipitated by a slightly-crazy Thrór who, desperate to reclaim some wealth and status after being driven out of Erebor, wandered into orc-infested Moria. In response Azog, self-proclaimed master of Moria, sent back Thrór's head with "AZOG" branded on the brow.
The resulting war, the War of of the Dwarves and Orcs, lasted for nine bloody years. The decisive battle was the Battle of Azanulbizar, which is the battle that An Unexpected Journey dramatizes. The battle is adapted pretty faithfully; this was the battle where Thorin earned the name Oakenshield, for instance, and it does take place outside the Eastern entrance to Moria2.
One main difference, the death of Thrór, does in fact have some precedent in the text:
Soon Náin made a great stroke with all his strength that remained, but Azog darted aside and kicked Náin's leg, so that the mattock splintered on the stone where he had stoo, but Náin stumbled forward. Then Azog with a swift swing hewed his neck. His mail-collar withstood the edge, but so heavy was the blow that Náin's neck was broken and he fell.
Then Azog laughed, and he lifted up his head to let forth a great yell of triumph
With some fairly minor differences, and a different murdered character, this is very similar to Thrór's death-scene in the film version. It even has a similar effect for Azog, though much more drastic:
Up the steps after him leaped a Dwarf with a red axe. It was Dáin Ironfoot3, Náin's son. Right before the doors [to Moria] he caught Azog, and there he slew him, and hewed off his head.
So despite some differences, it's reproduced largely faithfully.
What happened to Azog's arm, anyway?
Here's a picture from (I believe - it's a Google find) Desolation of Smaug, that shows the result:
In this image, we can just make out that Azog's left arm has been replaced by a "prosthesis" (Read: a big metal hook). We don't have any information about when this happened, but it was obviously after the Battle of Azanulbizar. It doesn't seem like a stretch to say that this is the Orc's attempt to treat their captain's injury. May the Valar bless the unfortunate Orc who got stuck with that task.
This promotional statue of Azog from SDCC (Courtesy of theonering.net) gives a better view of the arm
What's more, it seems that Azog 2.0 comes with replaceable limbs. In The Battle of the Five Armies, he's sporting a nifty sword-arm (by which I mean he has a giant, sharp piece of metal jabbed through his stump4):
1 Because historians are so good at naming things5
2 The same gate the Fellowship ran out of after losing Gandalf in Fellowship
3 Billy Connolly's character in The Battle of the Five Armies
4 If he had been born today, I get the feeling that Azog would be really into bod-modding.
5 More accurately, Dwarf-historians are so good at naming things. The First Age had awesome names for things: "The War of Wrath", "The Battle of Unnumbered Tears", "The Battle of the Sudden Flame", "The Battle Under the Stars", "The Battle of the Thousand Caves", "The Glorious Battle". The Dwarves just have to make history boring, I guess