A redeye is an overnight airplane flight. If you're traveling to New York from Los Angeles, you could take a redeye that leaves at ten p.m. and arrives around seven a.m.
Because they're crossing time zones and will lose several hours, travelers heading east sometimes choose to take a redeye. Theoretically, you can sleep on the plane during a redeye and be wide awake when you reach your destination. In reality, though, they're called redeyes for a reason: the red-rimmed eyes you have after an uncomfortable, sleepless night.