P = I*V in electricity. In this case, it doesn't really matter because we can pick any time span we wish. Each basic lamp, heater, or motor "works" with the electrical circuit. Typically, an open-close switch is all that is required to provide or detach electricity from the source to the circuit. For example, a light switch turns off and on the power supply to the lights, so they cannot burn out even if the power is on.
However, many devices need more control than an open-close switch can provide. For example, a heat pump uses a compressor to send air through a coil, which cools the air before it is sent back into the room. The compressor must be able to turn on and off at will, but this requires a semiconductor switch rather than an ordinary switch. Also, some devices have several circuits they want to operate at different times. For example, a radio has a speaker output and a headphone output, both of which need to be on at once in order for sound to come out of the radio. A single switch cannot do this job; instead, each circuit has its own switch that can be turned on or off as desired.
The term circuit also refers to any path along which an electric current may flow. For example, in a three-wire system such as the one used by houses until about 1930, every wall box had three cables running through it: one black, one white, and one ground (or green).
An purposefully open circuit would be the circuit to the turned-off lights in the room. Because the switch is in the "off" position, which "opens" the channel via which power would ordinarily flow, there is no closed path available for energy to flow to the lights. They will not light up when the power is turned back on.
When a switch is on, does that mean that the circuit is closed? No, because even with the switch on the circuit may still be opened by any one of several things: wiring defects, damaged wires, broken poles of a dual-pole switch, etc. If any part of the circuit is open, power will not reach the lamp and it will not light up.
If you check the wiring diagram for your house and don't see any lamps mentioned, this doesn't mean that they are wired out. Lamps can be wired directly into a branch circuit or they can be included as a member of a larger group called an extension cord. In the former case, they need to be listed on the wiring diagram to determine how they're connected to the circuit. In the latter case, they're usually just shown as being fed from a single pole source (house wiring) and then split up into individual lamps during installation. For example, if there were four lamps attached to one extension cord, they would each get their own leg from the source side to their respective terminals.