The Coven Overthrown (10 Points)
Achievement Reward: Banner Shape (Kurastian)
Kill Maghda on Normal Difficuly

Also achievements for killing Maghda on:
Nightmare (Banner Accent: Scarabs)
Hell (Banner Pattern: Small Hearts)
Inferno (Banner Sigil: Maghda)

Maghda is found in Act II during the "City of Blood" quest.

Murderer, Torturer, Monster (10 Points)
Achievement Reward: Banner Shape (Grim)
Kill Zoltun Kulle on Normal Difficulty

Additional achievements for harder difficulties:
Nightmare (Banner Accent: Key)
Hell (Banner Pattern: Large Waves)
Inferno (Banner Sigil: Zoltun Kulle)

Zoltun Kulle is in the Soulstone Chamber towards the end of Act II.

The Lord of Lies (10 Pts)
Achievement Reward: Banner Shape (Liberated)
Kill Belial on Nightmare Difficulty

Nightmare (Banner Accent: Axes)
Hell (Banner Pattern: Crossed)
Infernal (Banner Sigil: Belial)

Belial is found in the Imperial Palace in the City of Caldeum, at the end of Act II.

A Miner's Gold (10)
Complete the A Miner's Gold event

This event begins in the Howling Plateau. Talk to a miner standing near some elevators and help him recover a treasure chest to complete the event and earn this achievement.

Lair of the Lacuni (10)
Complete the Lair of the Lacuni event

This is in the Howling Plateau, just enter the Abandoned Mineworks (random dungeon) to start the event

The Rygnar Idol
(10)
Complete the Rygnar Idol event

Chamber of the Lost Idol (Random Spawn) in Stinging Winds.

The Restless Sands (10)
Complete the Restless Sands event

Speak to Mehtan the Necromancer in the Stinging Winds desert to start this event.

Guardian Spirits (10)
Complete the Guardian Spirits Event

Speak to a dying necromancer in the Stinging Winds to begin this quest

Crumbling Vault (10)
Complete the Crumbling Vault event

This is a timed event in the Stinging Winds in which you have to find the Vault Treasure Room before the ceiling collapses

Prisoners of Kamyr (10)
Complete the Prisoners of Kamyr event

This event can be found in the Dahlgur Oasis, you have to free some captive guards

Shrine of Rakanishu (10)
Complete the Shrine of Rakanishu event

This even, in the Dahlgur Oasis, involves destroying some summoning stones

Wheel of Misfortune (10)
Achievement Rewards: Banner Accent (Bow)
Experience all the possibilities of the Ancient Device in the Desolate Sands

This device has 10 possible outcomes, some lead you to random dungeons and some to encounters.

Lost Treasure of Khan Dakab (10)
Complete the Lost Treasure of Khan Dakab event

This is in the tomb of Khan Dakab, a random dungeon in Dahlgur Oasis

The Jewel of the East (10)
Explore the following areas of Caldeum:
Hidden Camp
Caldeum Bazaar
Searing Sands Inn
Gates of Caldeum
Sewers of Caldeum
The Wretched Pit
Flooded Causeway
Ruined Cistern
Imperial Palace
City of Caldeum

As you progress in the game, the story will take you to all the above locations, so no need to do anything special specifically for this game.

Just Deserts (10)
Explore the following areas of the Stinging Winds
Sunderred Canyon, Howling Plateau, Secret Altar, Hidden Conclave, Black Canyon Bridge, Black Canyon Mines, Stinging Winds, Khasim Outpost, Road to Alcarnus, Alcarnus, Lair of the Witch

Unparalleled Splendor (10)
Explore the following areas of the Dahlgur Oasis:
Path to the Oasis, Dahlgur Oasis, The Forgotten Ruins, Western Flow Control, Eastern Flow Control, Waterlogged Passage, Hidden Aqueducts, Ancient Path

The Bone Collector (10)
Explore the following areas of the Desolate Sands:
Vaulr of the Assassin
Cave of the Betrayer Level 1
Cave of the Betrayer Level 2
Desolate Sands

Navigating the Labyrinth (10)
Explore the following areas in the Archives of Zoltun Kulle:
Archives of Zoltun Kulle
The Terminus
The Unknown Depths
The Storm Halls
Realm of Shadow
Soulstone Chamber

Nooks and Crannies (10)
Explore the following dungeons in Act II
IMPORTANT: While most of these are random spawns, they always appear in the same zone/area. See the random dungeons guide to learn where to look for them: Diablo III Act II Random Dungeons Guide
Hadi's Claim Mine, Abandoned Cellar, Tunnels of the Rockwurm, Sirocco Caverns Level 1, Sirocco Caverns Level 2, Chamber of the Lost Idol, Tomb of Khan Dakab, Tomb of Sardar, Cave of Burowing Horror Level 1, Mysterious Cave Level 2, The Veiled Treasure, The Forgotten Ruins, The Fowl Lair, The Ruins Level 1, The Ruins Level 2, The Crumbling Vault, Vault Treasure Room, Vile Cavern Level 1, Vile Cavern Level 2, Flooded Cave Level 1, Flooded Cave Level 2, Alcarnus Cellar,  Sandy Cellar, Town Cellar, Abandoned Mineworks, Fuad's Cellar, Rotting Cellar, Ransacked Cellar, Storage Cellar, Old Fisherman's Cellar, Storm Cellar, Swampy Cellar, Deserted Cellar

Wanderer of Caldeum (10)
Achievement Rewards: Banner Accent (Horses)
Complete the Act II Exploration Achievements listed below:
The Jewel of the East
Just Deserts
Unparalleled Splendor
The Bone Collector
Navigating the Labyrinth
Nooks and Crannies

Judge of Character (10)
Read the following people lore books in Act II:
Fuad's Journal, Zoltun Kulle, The Musings of Sardar, The promises of Fezuul Al-Kazaar, Eirena's Journal, Iron Wolf Captain's Journal, The Legend of Zei the Trickster, Kala's Poem, The Last Will of Khan Dakab, Larra's Diary, Kulle's Journal Part 1, Kulle's Journal Part 2, Kulle's Journal Part 3, Kull's Journal Part 4, Leah's Journal Part 8, Leah's Journal Part 9, Leah's Journal Part 10, Leah's Journal Part 11, Leah's Journal Part 12, Andariel the Maiden of Anguish, Azmodan, the Lord of Sin, Baal the Lord of Destruction, Belial the Lord of Lies, Diablo the Lord of Terror, Duriel the Lord of PAin, The Seven Lords of Hell, Mephisto Lord of Hatred, A Love Letter, Necromancer's Log, The Imperial Guard's Orders, Crumbling Journal, Poltahr's Notes, The Ancient Portal Device, Blackened Journal, A Servant's Journal

Eavesdropper (10)
Read the following quest lore books in Act II:
Guard's Orders, Proclamation from the Imperial Guard, Missive to Maghda part I, Missive to Maghda Part 2, Deceiver's Orders Part 1, Deceivers Orders Part 2, Deceiver's Orders Part 3, The Feared Hero, Hunter's Journal Part 1, Hunter's Journal Part 2, Hunter's Journal Part 3, Hunter's Journal Part 4, Hunter's Journal Part 5, Hunter's Journal Part 6

Historian of Caldeum (10)
Read the following World Lore books in Act II
The Desert Aqueducts Part 1, The Desert Aqueducts Part 2, An Abridge History of Caldeum Part 1, An Abridge History of Caldeum Part 2, An Abridged History of Caldeum Part 3, An Abridged History of Caldeum Part 4, An Abridged History of Caldeum Part 5, An Abridged History of Caldeum Part 6, The Dahlgur Oasis, On the Desolate Sands Part 1, On the Desolate Sands Part 2, On the Desolate Sands Part 3, The Chamber of Faces

Beastmaster of Caldeum (10)
Achievement Rewards: Banner Sigil
Read the following Bestiary Lore Books in Act II:
The Accursed, Deceivers, Dune Dervish, Dune Thresher, Dust Imp, Electric Eel, Fallen Grunt, Fallen Lunatic, Fallen Shaman, Lacuni, Sand Dweller, Sand Wasp, Skeletal Guardian, Stinging Swarm, Blood Hawk, Rockworm

The Black Soulstone (10)
Achievement Reward: Banner Shape (Archer)
Complete all quests in Act II:
Shadows in the Desert
The Road to Alcarnus
City of Blood
A Royal Audience
Unexpected Allies
Betrayer of the Horadrim
Blood and Sand
The Black Soulstone
The Scouring of Caldeum
Lord of Lies