What exactly does 'Protection' enchantment do?

Protection will stack on different armor pieces, and provides comprehensive protection against all forms of damage except The Void and server kill commands. It is true that falling, fire, lava, poison, and even mob damage will be reduced, but false that the armor durability will always reduce more than normal. When falling off a building with Protection IV boots, the boots will not suffer any damage, producing the same results as feather falling III boots. The Armor will take more damage from Mobs and Players however.

It is possible to get feather falling IV AND protection IV boots, reducing fall damage immensely. All other pieces of armor with protection will add additional damage reduction in the same way the boots would. Protection appears to be one step down from specific defenses (ex: Protection III = Feather Falling II when taking fall damage, or Protection III = Fire Protection II when in lava).

TL;DR- Protection enchantments are slightly less effective than specific ones, but protect against everything at once. The best possible armor would be:

Helmet: Respiration III, Aqua Affinity I, Protection IV, Thorns III
Chest and legs: Protection IV, Thorns III
Boots: Protection IV, Feather Falling IV, Thorns III

Note that Only highest Level of Thorns take effect if present on multiple peices of armor. Source

What they mean by "Converts environmental damage to armor damage" is that it takes some of the damage dealt to your health (falling, damage from mobs, fire, TNT, etc.) and puts it into your armor durability instead, giving your armor more work and your health less damage.

NOTE: There are also Protection enchantments that are focused on one specific way of gaining damage.

Source: http://www.minecraftwiki.net/wiki/Enchanting