Cominius is one of Coriolanus' pals. Like a lot of the patricians, he's an important politician and also a respected military leader. He's one of Rome's consuls (which is kind of like being a president) and he's also chosen to lead the army against the Volscians when Rome invades the city of Corioles (1.1).
Here are two things to know about Cominius: he's loyal to Rome, and he also likes being at the top of the city's political and social food chain. That's why he's so worried about what could happen when the plebeians go into riot mode and threaten the social hierarchy. When the plebs get ready to throw down at the marketplace, Cominius warns "That is the way to lay the city flat; [...] In heaps and piles of ruin" (3.1.256, 259). Obviously, Cominius is a level headed guy who doesn't want to see Rome destroyed by civil warfare. At the same time, he seems worried that the plebeians could cause Rome's social hierarchy to crumble "in heaps and piles of ruin" as well. '
In other words, Cominius is all about maintaining the status quo. (He'd probably get on well with Valeria.) But he's not always successful at peace-making. When Coriolanus joins Rome's enemy (the Volscians) and prepares to destroy the city, Cominius tries to talk some sense into his old friend. Unfortunately for Coriolanus, he fails (5.1).