Modern derailers are spring loaded, pulled one way by the spring and the other way by the control cable. A «low-normal» derailer is one in which the spring pulls it toward the lower gear(s). If you release the tension on the cable, it will shift to the lowest gear.
Up until the late 1950s, all spring loaded derailers were low-normal type. When Campagnolo introduced the parallelogram-type rear derailer, they changed to high-normal, and most rear derailers made since then have been of the high-normal type.
The major advantage of high-normal rear derailers is that, when used with a low-normal front, both levers move in the same direction for double shifts. This makes it easier to perform a double shift with down-tube shift levers.
The major advantage of low-normal derailers is that they generally downshift a bit better than high-normal units.
Since the late 1990s, Shimano has attempted to revive the low-normal rear derailer design, using the trademark «RapidRise.» This has met with increasing acceptance by cyclists.
Sun Tour used to make high-normal front derailers. The principal advantage of this was that the front and rear shift levers moved in the same direction to either raise or lower the gear, which was less confusing for beginner cyclists.