N type semiconductor material has mobile electrons; P type semiconductor
material has a nearly-full band of electrons, which form a kind of log-jam: the
only mobility is due to the occasional voids in the band, called 'holes'.
Movement of an electron, from conduction band in an N region to a P region, is current induced in the LED by the battery (or other power supply).
That newly arrived electron is in the unstable state of being in
a high energy band while vacancies in the lower band (holes) are
prevalent. When it drops from the conduction band into a hole in the valence band, that's called 'recombination'. By conservation of energy, we
can expect to see a single photon that can escape the material as light (of a hue that represents the energy difference in those bands).
As to how the electron gets uncombined, that happens at the wire connection
(the positive terminal of the LED) as a steady flow of negative charge
goes into the wire and through the generator/battery/power source, into
a second wire, which connects to the negative terminal of that LED.
The wire-to-semiconductor connections are just as eventful as the light-emitting
PN junction, in their own way, but without emitting light. That is because the
large number of filled states, and large number of vacant states, in the
conduction band of a metal, allow for many thermal-energy-sized energy hops instead of one big leap across a bandgap.