Social Status was extremely important in Roman Society. Although some legends arise from, for example, Auxilary Soldiers who through extremely brave acts were given citizenship and status, most of it was short lived. Those of influence such as Marcus Crassus and Julius Caesar went great lengths to extend their social status. While in Gaul, Caesar had the Forum built. A marketplace that dwarfed the other buildings in Rome and was designed to remind those in Rome that he was ever present even whilst away. It worked, and caused the Senate to debate how powerful a man like Caesar, with his status, could become. Affairs and common controversy could negatively impact the social status of the elite unless tackled accordingly which led to 'mishaps'. This demonstrated how seriously social status was taken.

Yes. Very important. like most civilisations in most places status was a huge deal. Hierarchy was often overlapping and people could have many different shades of social status. The first was the distinction between slave and free. A free man was always higher up the hierarchy than a slave, he was not property, he could own property, he could marry and so on.

The next distinction is between citizen and non citizen. While in Athens a person might be a big cheese, in Rome they were a non citizen alien and thus, despite what status they might have back home in many legal aspects they were inferior to the bum in the street who happened to be a citizen. Incidentally, Roman clients could acquire a limited form of citizenship called the Latin Right which gave them many of the legal prerogatives of the citizen without the voting rights. The distinction between citizen and non citizen was essentially done away with by Caracalla in his eponymous edict of 212.

Another outmoded distinction was between patrician and pleb. Originally the Plebs had been barred from the highest office of Consul. But by the reign of Augustus much of this class distinction had evaporated and the wealthy plebs and patricians were extensively intermarried. The only importance that the distinction still had was that the powerful office of Tribune of the Plebs (which held a veto over all senate business) could only be held by a pleb. It is explicative that Publius Clodius gave up his patrician status by being adopted into a plebian family to run for the office of tribune of the plebs.

For a strictly wealth based criteria we have the distinction between equestrian and senator. If you property of more than 50k denarii you were an equestrian, the lower level of the aristocracy. This was increased to 100k by Augustus. To qualify for membership of the senate property in excess of one million sesterces was required.

After wealth we have two more distinctions, one based a combination of wealth and breeding. Julius Caesar, for instance, belonged to the Gens Julia, he was therefore of impeccable breeding and had many connections going back to the beginning of Roman history. However, he was not very rich, by the standards of the senatorial class. Just for context Pliny the younger counted himself as a "poor" senator with a fortune of 20 million sesterces and a slave work force of 500. A man with wealth but no breeding would find himself somewhat out of place in the Roman aristocracy. One of the ways that the concern manifested itself with was with marriage. A rich family which could count many members in high public offices would be desirable. A poor family with only one held praetorship three centuries ago would be much less desirable.

A further shade to this is the novus homo, the new man. These were men who rose from obscure origins and were the first in their families to hold high office. Examples include Cicero and Marius. Cicero, while he was an accomplished lawyer, orator and politician, rising to the consulship and defeating the Catalinian conspiracy, always felt slightly insecure about his equestrian origins.

In later times, in the elite, the old Roman aristocracy could be highly snobbish towards the provincial elites who were as well educated and as rich as they were. One example was Trajan, whose wealth came from a brick works in Spain, whose Roman ancestors had migrated to Spain as colonists. Cassius Dio for instance dismissed him as "an Iberian, and neither an Italian nor even an Italiot".

edit: Publius Clodius, not Tiberius Gracchus. Thanks /u/xenophontheathenian