It simply means measuring the cos phi and sin phi (or equivalenty the real and imaginary parts if you prefer complex notation) terms of the signal.
From which it follows that if you represent the signal using phasors, the quadratures (I and Q) correspond to the x and y axis, respectivly,
One obvious reason for measuring the quadratures is that you get BOTH the amplitude and the phase iof the signal, whereas "normal" homodyne detection will only give you the intensity.
In order to measure both quadratures in the MW regime you need a couple of mixers and a 90 degree phase shifter (in reality this will be a hybrid) or simply and IQ-demodulator.
If you are working at lower frequecies an ordinary lock-in amplifier will work (the "X" and "Y" outputs) .