The Paragon System is a new leveling system after you reach the maximum level in Diablo 3. So once you have reach level 60 you now have another 100 levels that take place. These levels grant bonuses such as, core stats like as Strength, Dexterity, Intelligence, and Vitality in similar amounts to what you would gain from a normal level while reaching level 60. As well as 3% Magic Find and 3% Gold Find.
This magic find and gold find totals up to 300% magic find and 300% gold find. This is the maximum percentage cap anyway, meaning there will be no need to have magic and gold find gear anymore, no more gear swapping during battles. This alone is a major improvement to the game.
Leveling up will be the same as before, the amount of experience needed to level up is unknown at the moment. This also being said every 10 levels your character icon in game will change as well. A good example of this can be seen below along with how the experience bar will look as well.