Exploration achievements (Under "General") reward you for exploring certain parts of the game world.

A Guiding Light (10 Points)
Achievement Rewards: Banner Accent
Use the Templar as your follower

The Templar is found during Act I, down at the Cathedral Level 3

Stolen Moments (10 Points)
Achievement Reward: Banner Accent
Use the Scoundrel as your follower

The Scoundrel can be found towards the second half of Act I, at the Fields of Misery, before getting to the Drowned Temple.

Rarely There (10 Pts)
Achievement Rewards: Banner Pattern
Equip a rare (yellow) item.

Yellow items have a random chance to drop from monsters or chests. You will probably find one on the Skeleton King who is the first major boss in Act I. If you want this achievement sooner, you can just go and buy the cheapest rare item from the Auction House.

Bless You (10)
Achievement Rewards: Banner Pattern
Get the following benefits from shrines:
Blessed (Reduced Damage)
Fortune (Increased chance to find gold and magic items)
Enlightened (Gain EXP)
Frenzied (Attack and Crit chance increased by 25%)

Shrines are found throughout the world, and in dungeons. They always provide a useful benefit so it's a good idea to get these blessings whenever you can regardless of the achievement.

Me, Myself and Eirena (10)
Achievement Rewards: Banner Accent
Use the Enchantress as your follower

You will find the Enchantress near the start of Act II, in the Sundered Canyon

Friends with Benefits (10)
Fully equip one of your followers

Check what slots your follower has available, and equip with any extra items you have.

Legends of the Brawl (10)
Achievement Reward: Banner Pattern (Large Lightning)
Equip a legendary (orange) item

The orange items are very rare and hard to find, but if you play long enough you will either find one, or be able to afford to buy it from the AH.

Socket to Me (10)
Socket a gem in an item

Iced Out (10)
Socket the following gem types:
Topaz, Emerald, Ruby, Amethyst

Chestmaster 2000 (10)
Open 2,000 chests.

Born to Dye
(10)
Achievement Rewards: Banner Accent (Potion)
Dye an item

Rainbow Connection(10)
Use the following dyes:
Infernal Dye, Abyssal Dye, Golden Dye, Spring Dye, Summer Dye, Autumn Dye, Royal Dye, Cardinal Dye, Mariner's Dye, Purity Dye, Aquatic Dye, Lovely Dye, Elegant Dye, Ranger's Dye, Winter Dye, Tanner's Dye, Forester's Dye, Desert Dye

Color Coordinated (10)
Equip your head, shoulders, torso, legs, feet and hands with items dyed the same color

Can be any color, but it has to be the same one for all items

Made it Work (10)
Achievement Reward: Banner Accent
Equip your head, shoulders, torso, legs, feet and hands with items dyed in all different colors

Make sure no two armor items you are wearing have the same color, or you don't get this achievement.

Wirt's Case Scenario (10)
Acquire Wirt's Cowbell

You can buy this bell from Squirt the Peddler in the Caldeum Market in Act II. This bell costs 100,000 gold so you will most likely not be able to afford it until later in the game.

Sets Appeal (10)
Complete one of the following level 60 unique item sets:
Immortal King
Inna's Majesty
Natalya's Wrath
Tal Rasha's Wrappings
Zunimassa's Spirit

Set items are colored green.

Space! I love Space (10)
Buy all stash upgrades

Stash upgrades let you store more items in your stash.

Best Dressed (10)
Equip a socketed item in the following slots:
Helm with Socket
Chest Armor with Socket
Leg Armor with Socket
Weapon with Socket
Amulet with Socket
Ring with Socket

Slice of Life (10)
Achievement Reward: Banner Accent
Use 50 health wells

I got what you need (10)
Acquire a Templar Relic, an Enchantress Focus, and a Scoundrel Token

Grave Buster (10)
Destroy 50 tombstones

The Weeping Hollow is a good area for this. If you run out of tombstones, just leave the game and come back to reset the area.

Market Research (10)
Meet the following wandering merchants:
Wandering Tinker: Tinker's Hovel (Sheltered Cottage) in the Fields of Misery in Act I
Rodger the Alchemist - Act I, Wortham Bluffs, near a bridge
Vendel the Armorsmith - Northern Highlands Watch Tower in Act I
Adenah the Curio Vendor - House of Curios in the NE part of Fields of Misery (Lonely Dwelling) in Act I
Kyr the Weaponsmith - Act I, highlands passage near the Halls of Agony
Zaven the Alchemist (Act II, Dahlgur Oasis, near a mysterious cave)
Larra - Black Canyon mines in a deserted Cellar (Act II)
Halmin the Alchemist (found in the Foundry, a random dungeon in the Battlefields in Act III)

Sword for Hire (10)
Earn 50,000 gold from quest rewards

Just keep doing the quests to get this one.

Haunted (10)
Meet the following ghosts
Lucious the depraved (Cemetery of the Forsaken, Act I)
Datamine (Defiled Crypt, Act I)
Drury Brown (Defiled Crypt, Act I)
John Gorham Coffin (Defiled Crypt, Act I)
Fecklar's Ghost (Festering Woods, Act I)
The Archivist (The Unknown Depths, Act II)
The Tomekeeper (Archives of Zoltun Kulle, Act II)

Shut up and take my money! (10)
Achievement Reward: Banner Sigil
Buy each type of item from merchants:
Sword, Mace, Mojo, Amulet, Belt, Boots, Bow, Bracers, Ceremonial Dagger, Chest Armor, Daibo, Crossbow, Dagger, Dye, Fist Weapon, Gloves, 1-Handed Crossbow, Helm, Leg Armor, Mighty Weapon, Wizard Source, Polearm, Quiver, Ring, Shield, Shoulder Armor, Spear, Spirit Stone, Staff, Wand