A command and control center (C&C) is a computer that controls a botnet (i.e. a network of compromised or zombie computers). Some botnets use distributed command and control systems making the more resilient.
From the command and control center, hackers can instruct multiple computers to perform their desired activities.
Command and control centers are often used to launch distributed denial-of-service attacks because they can instruct a vast number of computers to perform the same action at the same time.