A flight referred to as a red eye is an overnight flight. Due to the fact that they travel overnight, they are generally longer flights, with some of the shortest red eyes being from San Francisco to Chicago, or Denver to Washington, D.C. Except for flights that are extreme long haul flights (example: Newark to Singapore), you'll only find red eyes going West-to-East, as otherwise the change in time zones makes it so that the flight would either arrive after the destination airport closes, or before the departure airport opens. Note: this is the same reason that there aren't any flights from a West Coast city to an East Coast city in the late afternoon.
Red eyes exist mostly due to business travellers: since a West-to-East transcontinental flight otherwise takes up a large portion of a day, the day is wasted unless the traveller takes the red-eye. Although red eyes are generally cheaper then daytime flights, they are not always cheaper, as there is an extremely high demand for the flights, especially on longer routes, such as Honolulu to Dallas. Also, many red eyes are hub to hub flights, or flights from major cities to airline hubs, which leads to more demand and fuller flights. It is fairly rare to find an empty red eye.