The development of a higher class is essential to a feudal society's refinement. After its formation, citizens become more enthusiastic when building Wonders. The general populace also becomes somewhat more able to cope with overcrowding in big cities.
The main bonus of Aristocracy is the Wonder-building boost, and it should determine your strategy. Aristocracy is a stand-alone level 1 Policy, not connected to any other policies in the tree. This makes it the choice of the player when exactly to adopt it. If you want to build Wonders early, you can adopt it first of all Tradition policies; if not you may adopt it last to complete the tree. Or anywhere in between, when the right moment arises. The second bonus doesn't benefit you much in the early game (what city reaches 10 Population before turn 100?), but it becomes useful later on in the game.
Aristocracy is a form of government in which a group of the most prominent citizens rule. Membership to this group may be hereditary, or it may be through merit or wealth or all three. The term is derived from the Greek word aristokratia, meaning "the rule of the best." There is a very thin line between an aristocracy and an oligarchy, and which form a government actually is might very well depend upon one's point of view: the rulers might consider themselves aristocrats, while everybody else sees them as oligarchs.