A biosensor is a biochemical system that can detect and report the presence of a certain analyte.
A historical biosensor is the canary in the coal mine. These caged birds were used by miners to detect the presence of dangerous gasses. The birds are more sensitive to the gas, when the bird drops dead the miners could still get out in time.
With synthetic biology we are able to build very advanced biosensors that are able to sense very low concentrations of specific molecules. Each of these molecular biosensors consists of a sensor circuit that is able to detect the molecule and process the information that results in the activation of a reporter that translates the information into a detectable signal.