Background
In order for electric current to flow continuously, there must be a return path to the energy source (battery, mains). The energy source, outward path, other electrical devices and return path form a circuit. This is a closed circuit if there are no breaks or gaps, whereas if there is a break, this is an open circuit
Hence an open circuit is off, and a closed circuit is on. This must be very confusing for French students, as in French, the word for on is the same as the word for open (ouvert). Similarly, the word for off is the same as the word for closed (fermé).
Digital logic circuits
A signal is said to be open circuit if it is floating, i.e. not being pulled to the logic '0' rail or the logic '1' rail.
The 7400 series TTL chips read open circuit signals as high, i.e. logic 1, owing to the way in which TTL input circuits work. This is because the technology is based on bi-polar transistor technology, which requires current to work.
By contrast, more modern chips uses CMOS technology, which is voltage driven rather than current driven. As such, there is no implicit tying to either logic rail - you must do this yourself with a resistor. Failing to tie a floating, open circuit signal can result in spurious behaviour due to spikes of noise coming into the floating line. Also, the floating line could be subject to static discharge, which can destroy the chip.
Tri-state logic
If circuitry is in place that can detect the open circuit condition, this can be used to instigate a ternary system of logic: Logic 0, Logic 1 and open circuit. | 
