From Diablo Wiki
The Monster Power system was introduced into Diablo 3 in Patch v1.0.5 in October 2012, after being testable on the Diablo 3 Public Test Realm for several weeks. Inspired by the Diablo 2 system, Monster Power allows players to set a level from 0-10 which will increase the difficulty of the game. Patch 1.0.7 updated Monster Power to add considerably more experience per MP level, making for faster leveling up with higher MP levels.
With each level of Monster Power, the enemies increase in their hit points and damage. In exchange, players gain greater experience, a bonus to Magic Find and Gold Find, and the potential for bonus item drops.
The rewards and scaling values differ from Normal/Nightmare/Hell vs. Inferno, and playing on even MP1 in Inferno increases all Act 1 and Act 2 monster levels to 63, creating a big leap in difficulty and item drop potential, especially in Act One.
 Enabling Monster Power
To turn on Monster Power players need to open the Options menu and click the check box to enable it. After that all characters of any level, Hardcore and Softcore, can select their desired monster power on the game creation menu, next to the quest selection box, when creating a new game.
Monster Power is displayed on the screen in-game, so players can see what level it is set to when they join a game with MP enabled.
 Public Games
Monster power is now available and searchable in public games (via patch 1.0.7). You can open up your own MP game or join a game that is being played with MP via the public games menu.
 Monster Power Stats
Blizzard released full information about the Monster Power effects in mid-October 2012, detailing the exact changes triggered with each level of Monster Power with the aid of some infographics. Note that the v1.0.5 patch also contained a substantial nerf to overall Inferno difficulty, thus MP0 in v1.0.5 was considerably easier the game had been previously, and Blizzard estimated that MP2-3 was about the equivalent difficulty for pre-v1.05.
Player testing tended to disagree and at least in Act 1 and Act 2, the game felt harder on MP1 than it had been previously.
 Monster Power Pre-Inferno
One key detail is that Monster Power grants very different bonuses and changes while on Normal/Nightmare/Hell difficulties, compared to Inferno difficulty. For the first three difficulties, while characters are still leveling up and not yet finding end game quality gear, the following changes can be enabled.
Experienced players generally find playing a new, low or mid-level character on a high Monster Power setting to be very fun and a great way to level up more quickly than usual. 
 Monster Power on Inferno
Monster Power grants different bonuses and effects on Inferno, giving high end players a very challenging play mode by greatly increasing the hit points and damage of monsters. The values in the infographic below were accurate for the system when it was first released in patch v1.0.5, but a week after the patch went live Blizzard reduced the damage output. None of the other values were tweaked at that time.
The new values, which apply only to Inferno (Normal/NM/Hell were not adjusted) are:
- MP1: 109.60% (down from 114%)
- MP2: 120.12% (down from 130%)
- MP3: 131.65% (down from 148%)
- MP4: 144.29% (down from 169%)
- MP5: 158.14% (down from 193%)
- MP6: 173.32% (down from 220%)
- MP7: 189.96% (down from 250%)
- MP8: 208.20% (down from 285%)
- MP9: 228.18% (down from 325%)
- MP10: 250.09% (down from 371%)
Patch v1.07 considerably increased the experience gain per level, making higher MP levels more rewarding.
- MP 1: 25% (up from 10%)
- MP 2: 50% (up from 20%)
- MP 3: 80% (up from 30%)
- MP 4: 120%(up from 45%)
- MP 5: 165% (up from 60%)
- MP 6: 215% (up from 75%)
- MP 7: 275% (up from 95%)
- MP 8: 340% (up from 115%)
- MP 9: 420% (up from 135%
- MP 10: 510% (up from 160%)
A key factor for players new to the MP system is the fact that playing on even MP1 raises the monster level of all enemies in Inferno to 63. This changes nothing in Act 3 or Act 4, but on MP0, Act 1 = 61 and Act 2 = 62. Monsters at level 63 have much better odds to drop higher level gear, but are also much more difficult to battle in some ways, especially regarding their resistance to crowd control effects such as stun, chill, knockback, etc.
As you can see in the infographic above, there is no change to the item level potential of drops in Act 3 and 4, but with ML1+ monsters in Acts One and Two drop items with the same high level potential as Act 3 and 4.
 Chests and Barrels
In terms of Monster Power, gizmos will behave like monsters. At MP1+ in Inferno, gizmos in each act will be bumped up to level 63. (At MP0, nothing changes. They’ll just keep their normal level, like monsters do.)
Crafting, on the other hand, is not affected in any way by the Monster Power level.
 Player Reaction and Ramifications
After the system was implemented, players spent a great deal of time experimenting with and arguing about the ideal Monster Power level for item farming. After much early excitement over the increased experience and other bonuses, the conventional wisdom coalesced around the idea that fast runs on a lower MP setting are better than slower runs with a higher MP bonus.
The early consensus was that Act Three on MP0 was the easiest place for item farming, since the monsters were all level 63, and MP0 on Act Three is much easier than MP1 on Acts One or Two.
The bonus from Monster Power that players found impactful was the increased odds of finding a key from one of the Keywardens, or one of the Demonic Organs from the further battle with the uber bosses. Thus for key farming, playing on the highest MP possible, ideally while teamed up with other players for survival, is the recommended strategy.