An electronic circuit that processes binary data according to the rules of Boolean logic (AND, OR and NOT).  Officially an "integrated circuit," for the history of this world-changing discovery, see 
integrated circuit.  See 
chip and 
Bebop to the Boolean Boogie.
Digital Plumbing!
A digital circuit can be conceptualized as a mass of plumbing with water instead of electricity.  A transistor is a valve (open/close).  A resistor constrains the water into a smaller pipe.  A capacitor is a tank, and a diode ensures water flows in one direction.
Now imagine water passing through one valve that reaches a second valve, causing it to open and allowing water from a third valve to flow through that reaches a fourth valve, and so on.
From Logic to Plumbing
The plumbing diagram above depicts a circuit known as a "half-adder," which adds one bit to another in every computer.  To understand this at a gut level, look up 
half-adder and trace the path yourself with paper and pencil.