Internal reversibility

For a reversible process, the overall change in entropy is zero (i.e. no entropy is created). However, entropy can be transferred between different systems and the environment through reversible processes.

For a system transferring heat/work to/from its environment - as I understand it, for an internally reversible process, no irreversibilities exist within the system. This typically means that the system changes very slowly, without friction, through a series of equilibrium states. Conversely, internal irreversibilities occur due to friction or a system moving through non-equilibrium states.

An externally reversible process means that heat is transferred to the system by the surrounding environment very slowly, through a series of infinitessimal temperature differences. Heat that is transferred to the system across a finite temperature difference will cause external irreversibilities.

It is possible for a process on a system to be both internally and externally reversible. This link might help you to further visualize the differences.

An irreversible process is forgetful. Hence, an internally reversible process might be regarded as one whose initial state is 'remembered' throughout, whereas in an externally reversible process the initial state would be imprinted on surroundings in a way that could eventually be retrieved.