A small computer on a single integrated circuit containing a processor core, memory, and programmable input/output peripherals.
Microcontrollers are frequently used in many parts of a robot. A single microcontroller may implement one single function of a robot, (servo control, data collection), or may implement many functions. Some robots contain only one microcontroller which carries out all of the robot's sensing, computational and data handling tasks.
Most robotics enthusiasts use one "prototyping platform" PCB that includes a microcontroller, connected to one or more "breakout board" PCBs that include some kind of motor driver, which is in turn connected to the actual motors.
Robotics: Microcontrollers has a few more words about microcontrollers that are popular in robotics.
Robotics: Computer control has a few words comparing boards -- "protyping platform", "SBC", "multichip module", etc. -- that are popular in robotics, each one supporting one of the above microcontrollers.