Conformation identifying cis and trans
Draw the cycle as if it were planar (in fact it is not, but just imagine for a moment), with one hydrogen at each carbon pointing upwards with respect to the plane, and the other downwards. Now, if we introduce two substituents to different carbons on the same side of the plane, that would be the cis isomer, otherwise trans.

As for the stability of equatorial conformers, that's purely for sterical reasons. Draw the cycle, this time in its real geometry, and see what happens if we have two bulky groups in equatorial positions on the same side. Clearly they will be unpleasantly close to each other.