When the play opens, the plebeians are rioting for better access to the city's food supply. Basically, they accuse the patricians of hoarding all the grain while the common people starve. That's when a smooth-talking patrician named Menenius shows up and tells them a little fable to calm them down (1.1.87-163).
Need a quick summary? Here's how it goes:
One day, the members of a body rebelled against the belly and accused it of being lazy and hogging all the food while the rest of the body parts did all the work and got none. (Duh, body parts can totally talk.) The belly replied that, sure it's true that the belly is the body's "storehouse" for food but it's also a distribution center because the belly sends out nutrients through the bloodstream to all the other body parts.
So, what's Menenius' point? Rome is just like a human body. The Senators are the "belly" because they're the ones in charge of collecting the city's grain before dispersing it to the common people of Rome, who are the "body parts." At one point, Menenius even says that the guy over there with the big mouth is like a grouchy big "toe" (1.1.155). In other words: quit complaining, because they're not really starving and the ruling class is actually taking good care of them, if they were just smart enough to know it. If the plebeians (body parts) continue to revolt against the belly (the Senators), the whole body (Rome) is going to die.
If you're a fan of Aesop's Fables, you may have heard some version of this story before. It turns out that this is actually a famous political metaphor that was hugely popular in Shakespeare's day. Literary critics and historians even have a fancy name for it--they like to call it the "body politic" metaphor, which just means that it's a metaphor that compares a city (like Rome) or a country (like England) to a human body. The metaphor was often used by political leaders to support the idea of social hierarchy and to try to keep the common people in check.
Shakespeare liked it, too. In Titus Andronicus, he litters the stage with hacked up body parts to show that Rome has been torn apart by civil conflict. The metaphor also shows up in King Lear, where Shakespeare fills the play with references to sexually transmitted diseases to emphasizes the idea that Lear's kingdom is "sick."
In Coriolanus, Rome is definitely sick. Of course, the patricians blame the unruly plebeians for Rome's problems. When the plebs riot over food shortages, Coriolanus shows up and calls them a bunch of "scabs" (1.1.166). Later, he says they are "measles" that "tetter" or disfigure the body's skin (3.1). He also accuses them of being a "multitudinous tongue," licking up poison that will destroy Rome (3.1.156). (Gross.) Big shock, the plebeians and their tribunes see things differently. To them, Coriolanus is the disease that plagues Rome's political body, a gangrened foot (3.1.305) that "must be cut away" (3.1.293).
It seems pretty clear that Shakespeare is telling us that Rome's political state isn't exactly healthy—but we'll let you decide who's to blame.