Effective area or effective aperture of antenna
If I understand your question correctly, you are looking for the Polarization Loss Factor (PLF). As you already assumed, antennas with different polarizations will suffer a loss according to the PLF.  

The PLF is defined for linear polarization by the rotation angle φ between the antennas as PLF = cos²(φ), hence the most extreme case occurs with a difference of 90°/270° (one antenna with vertical and one with horizontal polarization), where in theory total signal loss will result. Sidenote: In reality you may still be able to receive a signal, but with a loss of 20dB or more.  

With circular polarization total signal loss (or >20dB) will occur if the antenna is designed for one direction and the field is rotating the other direction, for example a Left Hand Circularly Polarized (LHCP) antenna with a Right Hand Circularly Polarized (RHCP) field.

If you use a linear polarized antenna with a circular polarized field or the other way round, the loss will be 3dB, no matter what the rotation is. This is because circular polarization consists of two orthogonal linear polarized fields 90° out of phase and the linear polarized antenna will only receive the in-phase component.  

For your last question about the importance of effective aperture in a receiving antenna, remember the reciprocity property of antennas: Transmit and receive properties of an antenna are identical and equally important for the signal transmission. You will get antenna gain (and aperture and polarization effects) for both antennas.