How long will my circuit run normally if its battery level reduces?
Some may, some may not, it depends on what components are in use.

You usually design a circuit so that it can operate over the range of voltages that the chosen battery or supply will deliver. 

Most ICs will specify a minimum voltage at which they will work in their data sheet. However, beware, even when an individual IC is 'working', some of its parameters, so current drive, bandwidth etc, may be less at a lower rail voltage than the typicals you hoped for. Your system design has to allow for the lower performance at extremes of voltage.

If you have a circuit that works only over a small range of voltage, power it from a regulator that can accept the voltage variation from the battery. A switch mode regulator will make best use of the energy in the battery.