From Blizzard's Item Guide: I do think they say this part best!
Magic find affects the quality of items you acquire from killing enemies (but not treasure chests, vases, weapon racks, or other environmental objects). It doesn’t give you a chance to get more items on a given kill, but instead it increases the chance that an item you find will be magical, and more potent than it would be otherwise.
When a monster drops an item, Diablo III randomly determines the item’s quality from a chart that includes item quality and the number of affixes present. The game randomly "rolls" on each property in the chart to determine which affixes your item will get. Your magic find score is applied as a bonus to these rolls.
For example, if a monster has a 4% chance to drop a 6-affix rare item and you have +50% magic find, it now has a 6% chance to drop that item.
Item quality is checked in the following order:
Using the above example, when your roll ‘misses’ a higher-level item quality, the item generator proceeds to the next lowest item quality in the chain (in this case, checking to see if you got a 6-affix rare, then checking for a 5-affix rare, and so on). Your magic find bonus applies to each roll. If the same monster has a 10% chance to drop a 5-affix rare item and you have 50% magic find, you now have a 15% chance to get a 5-affix rare item. (end quote - http://us.battle.net/d3/en/game/guide/items/equipment )
In Diablo III, the slot drop does not permit the lower level items to be generated, in order to ensure that items you can currently USE on your character will drop rather than useless gear you would have left behind or given away ages ago. An example:
Leoric's Signet is a level 14 ring. To find this ring, you must kill monsters that drop level 14 - 16 rings. They can be found in Act 2 normal, from the Festering Cistern (just before the Oasis) until you reach the Desolate Sands. This item can not be found ANYWHERE ELSE IN THE GAME, and also cannot be found in nightmare, hell, or inferno. To add to the frustration, this ring cannot be found past the Desolate Sands because level 17 rings start dropping in the dungeons there. If you roll a unique ring and it's level 17, you have found a Manald Heal, which is a totally different unique ring.
Adding Magic Find
Adding Magic Find (or MF) to your character aids you in finding higher quality items as you adventure. In and of itself, it provides no useful benefit, and can take up a slot on your equip that could be better put towards keeping your character alive or allowing you to dish out more pain to those evildoers.
Diablo III adds additional features to what we know and love of Magic Find, first and foremost being so many ways to obtain it. You can find it on your equipment, apply it from shrines you find, Add stacks of Nephalem Valor, add Monster Power to increase it, play with more players to get a little more, increase your Paragon Level, and even socket your helmet with a Topaz to get just that little bit more ... But Diablo III also introduces the concept of a Magic Find cap, rather than diminishing returns, which plagued players of Diablo II. Your combined equipment (plus 20% of your follower's magic find) and your Paragon Level cannot exceed 300% MF. All other effects can apply to increase your magic find over the cap.