Classical Rome was in many ways founded on a social inequality. Essentially, Roman society, at least free people, was fundamentally divided between plebeians and patricians.
Plebeians were common people who were initially denied political voice in republican Rome. "Plebs" contained many variations, and were by definition free, but they lacked the social status of patricians, who inherited their positions through their family line. Eventually, class tensions between the plebs and patricians led to the establishment of plebeian officers, the most senior of which were tribunes.
Patricians, on the other hand, were aristocrats, representing a much smaller percentage of Roman society (probably about 10 percent.) While their status did not necessarily depend on their wealth, they did tend to be large landowners. More importantly, they controlled politics in Rome, and used their power to get policies, such as taxation and debt laws, to benefit them.
There were many different classes of people in the Roman Empire.
Namely (from lowest to highest)
4. Equestrian class
5. Senatorial class
Women were also highly unimportant figures in Roman society. There is no existing text found that was written by a woman which proves this. However there were a few exceptions. For example, in Pompeii, the cloth workers guild (a major and large business) was run by a woman.
This is of course just a vague outline of the classes. rrteacher's answer holds more detail.