Strain gages are connected to wheatstone bridge network in order to measure the strain. The basic wheatstone bridge has 4 resistances connected end to end, called arms of the bridge. Of these 4 arms, either one or two arms are strain gages, which actively change resistance, and are known as quarter bridge or half bridge configurations. The other remaining arms are made of precision resistances, usually inside the measuring instrument.
In some cases, all 4 arms of the bridge are strain gages actively participating in strain measurement - this is called full bridge configuration. In all cases the wheatstone bridge is completed to have 4 arms.
A single strain gage as a !/4 bridge is most commonly used in structural testing, to reduce cost - but strain gages are sensitive to temperature changes , lead wire resistances, and other effects that introduce error in strain measurement. Two strain gages in 1/2 bridge configuration is useful in cancelling out temperature and other effects , minimizing the errors, and also increasing the output of the bridge to give a stronger signal.
For most transducer applications, 4 strain gages in full bridge configuration are used. This helps in cancelling out temperature effects, and increasing stability and sensitivity of the wheatstone bridge.

The method of completing the bridge for 1/4, half and full bridges is described in this video presentation.