The principle can be shown by imagining a giant stationary pendulum, with a stylus on the bottom. As the earth starts to move, the heavy mass of the pendulum has the inertia to stay still in the non-earth frame of reference. The result is that the stylus scratches a pattern corresponding with the earth's movement.
Modern instruments are, of course, electronic. These seismometers resemble accelerometers, and have become all-encompassing instruments. In previous years, seismometers would 'clip' or go off-scale for ground motion that was strong enough to be felt by people. At this point, the only instruments that would work would be the less sensitive accelerometers.
The modern broad-band seismometer (so called because of the capacity to record a very broad range of frequencies) consists of a small 'proof mass', confined by electrical forces, driven by sophisticated electronics. As the earth moves, the electronics attempt to hold the mass steady through a feed-back circuit. The amount of force necessary to achieve this is then recorded.
Accelerometers output this directly as acceleration (remembering Newton's F=ma), but seismometers use an integrating circuit to output velocity.
Seismometers spaced in an array can be used to precisely locate, in three dimensions, the source of the earthquake, using the time it takes for seismic waves to propagate away from the epicentre, the point of fault rupture. Seismometers are also used to detect underground nuclear test explosions.
Excellent history of seismometers " class="external">http://neic.usgs.gov/neis/seismology/history_seis.html Nice Java applet of old damped-mass seismometer http://www.ifg.tu-clausthal.de/java/seis/sdem_app-e.html