Aristocrats are those of higher society that almost always hold not only money and power, but a "noble" bloodline as well. They are often found at the top of societies, running them for better or for worse, and maintain a large influence on the world despite their smaller numbers than laymen.
As their group title implies, royalty are those who belong to a royal bloodline. They are the rulers or other such leaders of their land and territories. Royalty ranges from kings who govern over entire countries, to small tribal chieftains that many only hold a few miles of land under their influence. Members of royal families hold high, if not total, influence and control over their realms and are the head of the government of most places, whether it be actual power or simply serving as figureheads to other individuals that hold the real power behind their thrones.
Though not quite royalty (but sometimes related to royalty), nobles are nonetheless powerful and high-ranking folk in society. Within the noble class are various subgroups in most cultures, ranging from barons to dukes and other ranks. Many nobles hold lands and some governmental influence themselves, but still others are only noble in name and in purse (and some, not even in that). They are the group most common to purchase slaves in the territories where it is still legal (and some places where it is not), but not all nobles approve of the trade of persons. Like royalty, they can be corrupt and destructive, kind and benevolent, or somewhere in between.
Officials are those at the bottom or just below the rank of nobility, but often still above the layman class. They serve often as more local governmental representatives, such as governors and policemen. Certain officials, such as ambassadors, serve higher and more important roles and are sometimes distant relatives of royalty, or rarely, royalty themselves. Many officials, especially those in looser and smaller governments or tribal areas, are of the commonwealth but still hold more power than their neighbors to help uphold the law.