What is the difference between an internal and an external circuit?

An internal circuit is a circuit within a conductor. An external circuit is a circuit within wires and devices.

Note: In the United Kingdom, its defined differently. The internal circuit is the circuit inside the supply (perhaps a cell or powerpack).  

The internal circuit consists of two components, an emf source and internal resistance.  

The external circuits is everything that is connected between the terminals of the power supply.  This includes all of the wires, resistors, lamps, capacitors etc.