The first denizen of common sense to fall victim to the was the ``obvious'' notion that if two physical events occur at the same time in my reference frame, they must occur at the same time in any reference frame. This is not true unless they also occur at the same place. Let's see why.
Einstein was fond of performing imaginary experiments in his head -- Gedankenexperimenten in German -- because the resultant laboratory was larger than anything he could fit into the patent office and better equipped than even today's funding agencies could afford. Unfortunately, the laboratory of the imagination also affords the option of altering the Laws of Physics to suit one's expectations, which means that only a person with a striking penchant for honesty and introspection can work there without producing mostly fantasies. Einstein was such a person, as witnessed by the ironic fact that he used the Gedankenexperiment to dismantle much of our common sense and replace it with a stranger truth. Anyway, one of his devices was the laboratory aboard a fast-moving vehicle. He often spoke of trains, the most familiar form of transportation in Switzerland to this day; I will translate this into the glass spaceship moving past a ``stationary'' observer [someone has to be designated ``at rest,'' although of course the choice is arbitrary].
Figure:
A flash bulb is set off in the centre of a glass spaceship O'
at the instant it coincides with a fixed observer O.
As the spaceship moves by at velocity u relative to O,
the light propagates toward the bow and stern of the ship at
the same speed c in both frames.
In Fig. 
both observers (O and O') must measure
the same velocity (c) for the light from the flash bulb.
The light propagates outward symmetrically in all directions
(in particular, to the right and left)
from the point where the bulb went off in either frame of reference.
In the O' frame, if the two detectors
are equidistant from that point
they will both detect the light simultaneously,
but in the O frame the stern of the spaceship moves closer to the
source of the flash while the bow moves away, so the stern detector
will detect the flash before the bow detector!
This is not just an optical illusion or some misinterpretation of the experimental results; this is actually what happens! What is simultaneous for O' is not for O, and vice versa. Common sense notwithstanding, SIMULTANEITY is relative.