Darkmoon Despoiler

Collecting all the Darkmoon Faire Artifacts listed below will give you this achievement. In addition, you will also get the following 2 achievements, Darkmoon Dungeoneer and Darkmoon Defender, while working on this one.

The full list of Darkmoon Faire Artifacts and the quests related to it are:

1. 
   
2. Fallen Adventurer's Journal & The Captured Journal
3. 
   
4. Banner of the Fallen & A Captured Banner
5. 
   
6. Captured Insignia & The Enemy's Insignia
7. 
   
8. Imbued Crystal & A Curious Crystal
9. 
   
10. Monstrous Egg & [quest=29448]/An Exotic Egg
11. 
    
12. Mysterious Grimoire & An Intriguing Grimoire/[quest=29449]
13. 
    
14. Ornate Weapon & A Wondrous Weapon/[quest=29450]
15. 
    
16. A Treatise on Strategy & The Master Strategist
17. 
    
18. Soothsayer's Runes & Tools of Divination/[quest=29465]
19. 
    

To begin gathering the artifacts, Blizzard has stated (Source):

In addition, the [item=71634] can be obtained from [npc=10445], who resides in the first tent on your left as you enter the Darkmoon Faire ground on Darkmoon Island.

The Darkmoon Faire Field Guide

It’s your passport to riches, my friend. Y’see, we need a few things -- just some junk, nothing valuable to a big hero like you. We call ‘em Darkmoon Artifacts, and there’s all different kinds to be found all over Azeroth. The Darkmoon Field Guide helps you discover artifacts while you’re explorin’ dungeons, slayin’ monsters, and fightin’ in Battlegrounds. Without a guide, you’d never notice ‘em, and it’ll help keep you focused on the stuff we want. Whether you just reached level 10, or you’ve crushed the biggest baddies in the land, we need somethin’ from nearly everybody. You won’t have to go too far out of your way to get ‘em and each month you’ll get a new opportunity to seek out an Artifact for us. What do you get out of the deal? Don’t you worry, you’ll get your cut. When you bring a Darkmoon Artifact back to us, you’ll get valuable experience (it builds character, you know!), earn a better reputation, and possibly earn precious Darkmoon Faire Prize Tickets too!

Updated for mop and it looks like there are no drops in [zone=5918]. According to the wowhead, A Treatise on Strategy drops in heroic mode, and the other items drop in normal modes off the respective boss. I.e.: Ornate Weapon drops in normal and not heroic.

I've compiled a easy to see table of where each item for Darkmoon Dungeoneer can be found to get the most bang for your regular/heroic dungeon queues since Blizzard "fixed" the drops so that [faction=909] monthly quest items can only be looted in level appropriate dungeons. I'm keeping the level 80 and 85 data for people who have not upgraded from the Battlechest and Cata to MoP.

Notes:
This is ordered in alphabetical order, with the instances that drops the most items for DMF items towards the top.
If you find that a boss drops something not listed here. Leave a reply and I'll add it in, eventually.

Soothsayer's Runes drops from the first bosses in every raid at level.

Fallen Adventurer's Journal, Banner of the Fallen and Captured Insignia can be looted only while in a battleground.

Mists of Pandaria

Imbued Crystal

Monstrous Egg

Mysterious Grimoire

Ornate Weapon

A Treatise on Strategy

[zone=6182]

[npc=61398]

[npc=61398]

[zone=6052]

[npc=58632]

[npc=58632]

[zone=6109]

[npc=3977]

[npc=59223]

[npc=59223]

[zone=6066]

[npc=59080]

[npc=59080]

[zone=5963]

[npc=56717]

[npc=56717]

[zone=5956]

[npc=56439]

[npc=56439]

[zone=5976]

[npc=56636]

[zone=5918]

Cataclysm

Imbued Crystal

Monstrous Egg

Mysterious Grimoire

Ornate Weapon

A Treatise on Strategy

[zone=4945]

[npc=39732]

[npc=39425]

[npc=39378]

[zone=5844]

[npc=54938]

[npc=54968]

[npc=54938]

[zone=5396]

[npc=43614]

[npc=44577]

[npc=43612]

[zone=4926]

[npc=39698]

[npc=39705]

[zone=5789]

[npc=54432]

[npc=54432]

[zone=4950]

[npc=40177]

[npc=39625]

[zone=209]

[npc=46964]

[npc=4278]

[zone=5035]

[npc=43875]

[npc=43878]

[zone=5004]

[npc=40586]

[npc=40765]

[zone=5788]

[npc=54853]

[npc=54969]

[zone=1581]

[npc=47626]

[zone=5088]

[npc=43214]

Wrath of the Lich King

Imbued Crystal

Monstrous Egg

Mysterious Grimoire

Ornate Weapon

[zone=4494]

[npc=29310]

[npc=29308]

[zone=4196]

[npc=27483]

[npc=26631]

[zone=4265]

[npc=26794]

[npc=26723]

[zone=4277]

[npc=29120]

[zone=4416]

[npc=29306]

[zone=4272]

[npc=28923]

[zone=4264]

[npc=27975]

[zone=4813]

[npc=36658]

[zone=4100]

[npc=26530]

[zone=4415]

[npc=31134]

[zone=1196]

[npc=26693]

Please keep in mind that the quests are not repeatable during one DMF festival. After you complete the quest you cannot get it for the second time, as it states when you use an item for the second time "You have already completed this quest". In other words - you can do the quest once every month.
Now, to all the naysayers, Blizzard has confirmed that these quests are to be repeated each month. Source
They have also stated that the quest drops that happen in the dungeons/raids/BG’s happen all the time, even if the faire isn’t in town. Source