When does electric current flows?

Answer 1

Simple Answer:

The definition of a current is a collection of moving charges.

Hence for current to flow, one needs charge and one needs it to move.

Electric current flows when charge move - that is the definition.

Long Answer:

One generally sees current flow on Earth with a conductor and a voltage or potential difference that creates and electric field and charges in the conductor feel a force and move creating current.

The conductor may be a wire and the charges may be electrons.

The conductor might also be water with dissolved salt and the electric field causes the salt ions to move and it is the moving ions that constitute the current.

If you are the Sun and you are producing electrons and protons on the solar wind, then complex electromagnetic fields may have ejected the charges from the sun, but the current that is the charge moving through space is maintained just by the inertial of the particles. You don't need an electric field to maintain the current and you don't need a conductor to hold the charges.

For a current to flow, you need net charge moving. That is all. Current is technically charge density times velocity.