A feature of some MIDI-equipped synths which allows two or more units of the same model to be ganged, via MIDI cables, to increase available polyphony. Consider an example: a six-voice polyphonic synth is playing a six-note chord. The performer plays a seventh note (or a seventh note is received via MIDI). Instead of stealing a voice from a sounding note, or ignoring the seventh note, the synth sends a Note on message out of its MIDI Out port, to a second synth (presumably the same model), which receives the note via its MIDI In and plays the note. When the note is subsequently released, the first synth forwards the Note off message for that note to the second synth.
Generally, a synth that implements MIDI Overflow will send the second synth all control-type information that it receives (Pitch wheel, Program change, Continuous controller, etc.) so that the two synths remain in sync timbrally. Sysex can be problematic, depending on exactly what the message does.
MIDI overflow should not be confused with MIDI thru.