In the 8th and 7th centuries BCE, the Roman Patricians were a small, oligarchic class made up of a few powerful families. Patrician status was determined by birth, not authority or wealth. Those who were members of the state but were not Patricians were called Plebeians. The social distinction between the two slowly faded over time. By the time of the Late Republic, many Plebeian families had become entrenched senatorial families with wealth and many famous ancestors. On the other hand, some patrician branches of families had become impoverished and had not had politically successful members in centuries. By the time of Augustus, the only real distinction was tradition: they could trace their lineages back to the early days of Rome (supposedly) and could feel good that they belonged to a very small and special group. There were many more plebeians than patricians, of course, and so the notion of plebeians as rabble both persisted and was technically accurate, statistically speaking. But the senate was stuffed with plebeians and had been for centuries. Because "patrician" was no longer a very useful word for denoting aristocratic status, the term nobilis arises. The nobiles were those, patrician and plebeian, whose families had achieved greatness in the past in the Roman system (via war, politics, etc). Most nobiles were senatorial families. There was another class, the equites or "equestrians" or "knights" (in archaic English writing), who were roughly equivalent to the bourgeois. They were frequently wealthy, sometimes grossly wealthy, but did not always have a long history of election success (=senatorial service) in their families. Many were "Italian" instead of Roman before the Social War in the 1st century. Though this generalizes, we often explain the equites as the "business class" of Rome--the ones who controlled the shipping corporations, the banking apparatus, the construction firms, the logistics of warfare, the tax-farming, etc. It is a confusing system.

Here are a few well-known Patrician families. Many of these would later have plebeian branches in addition (these families tended to get gigantic, and things get muddied via marriages over the centuries): the gens Aemilia; Claudia; Fabia; Cornelia; Horatia; Iulia; Lucretia; Mucia; Sempronia; Sergia; Servilia; Sulpicia; Tarquinia; Valeria.

Many familiar names from the Late Republic were from plebeian families: Gaius Marius was plebeian, but his rival Sulla was a thoroughbred patrician (Lucius Cornelius Sulla); Pompey (gens Pompeia); Cicero; Marc Anthony (Marcus Antonius). Famously, the gangster Clodius had arranged to have himself moved into the "plebeian" branch of the family in order to be elected a tribune of the plebs; he had originally belonged to the gens Claudia (the spelling difference reflects a change in the way the word was pronounced over the centuries, with the Claudii wishing to retain the old-timey pronunciation).

Plebeians were the poor or the working class

Plebeians were anybody who was not a patrician. A patrician was someone descended directly through the male line without having suffered loss of status somewhere along the line from one of the "patres," the 100 legendary direct advisors to Romulus who supposedly made up the first senate. Patricians tended to be wealthy, mostly because it was a social expectation of the order, but there was no wealth qualification for patrician status, it was purely hereditary.

Access to the senate and to elected magistracy had been opened up to plebeians in the early fourth century, and by law at least one consul had to be a plebeian. There's some dispute over whether plebeians were ever barred from either role, as Livy records a rather large number of consuls and senators early in the Republic who must belong to plebeian families, including probably L. Brutus himself. In any case the senatorial order--the only "aristocracy" worth speaking of in the truly historical period--was made up predominantly of men of plebeian rank by Augustus' time. Of about 50 or so patrician families that we know of through Livy and others in the earliest period of the Republic, only 14 can be firmly shown to have had surviving patrician lines during Caesar's lifetime. Of these several families, like the gens Quinctia, were no longer politically active. The vast majority of the great political figures of the first century BC are all plebeian in rank. Not that that distinction meant very much. Following the early fourth century the only distinction that mattered was whether one had senatorial status or not.

The confusion likely arises from the fact that Livy makes a big deal of the Conflict of the Orders between patricians and plebeians, a political struggle that is supposed to have happened in the very earliest Republic but which was mostly over by the end of the fourth century. The Conflict of the Orders was part of the tradition, but authors before Livy make very little reference to it even though it takes prominence in Livy. Particularly in the nineteenth century when Livy started to become especially popular a lot of non-specialists relied on him heavily, without really realizing a lot of the problems in doing so. Not least of these was Marx, who was criticized even by sympathetic voices for really badly misunderstanding Roman social structure (and we know now that his view of the Roman economy, which was a typical one at the time, is pretty much a fantasy), but there were many others. Livy gives the impression that plebeian-patrician conflict was the primary political conflict at Rome, and if that theme became less prominent over time we don't really know, because Livy's later books have all been lost. It's become commonplace in our society to carry those misinterpretations of Roman social structure from the nineteenth century through to our own time.