At the heart of diving is the Scuba Unit. The unit, in its simplest form consists of a gas tank, first stage regulator, pressure gauge, low pressure inflator, buoyancy device, and primary and redundant second stage regulators.
A common misconception is that divers use oxygen tanks. This is not true. In fact, breathing pure oxygen can be very dangerous for divers due to an effect called partial pressure. The gas in a diver's tank can vary greatly, but is usually between 21% and 36% oxygen, with the remaining volume occupied by nitrogen.
Early scuba units consisted of no more than a gas tank, first stage regulator to bring the tank pressure down to an intermediate pressure, and a second stage regulator to bring the intermediate pressure down to ambient for the diver to breath without difficulty. This was usually paired with a wrist mounted depth gauge and timer. Early divers made use of a "J-Valve" on their tanks. The J-Valve kept a small portion of the diver's breathing gas in reserve and would allow the diver to use it for ascent after the diver pulled a lever. The diver would feel a resistance when breathing at the end of the dive which would signal him to pull the J-Valve and open up his reserve.
The modern scuba unit is much more safe than the units of early divers. The J-Valve was replaced by a pressure gauge to accurately tell the diver the amount of air he has left; a second second stage (and sometimes another first stage as well) was added so that in case of an emergency, there is a backup available; modern computers keep track of a diver's vital dive information.
In addition to these changes, the quality of the equipment has gone up considerably as the sport has evolved. Dive Theory, the science behind diving, has also progressed quite a ways since the early US Navy experiments.