The boost tells you the amount of pressure inside the air intake system that is being generated by the turbocharger. In a normally aspirated (ie, no turbo or supercharger) the air (in a diesel) or air/fuel mixture (in a gasser) is "sucked" into the cylinder by the low pressure created when the piston travels down in the cylinder in the intake stroke. This creates a vaccuum (negative boost) condition inside the intake system. Think about what happens when you expand your lungs to create negative pressure when you drink from a straw.
Naturally, you can put more air into the cylinder in the same amount of time if the air is already under pressure when the intake valve opens. This is the purpose of the turbocharger. The turbo is basically two fans with a solid axle between them. One "fan" is in the exhaust stream, the other is in the intake stream. As the engine runs the exhaust rushing past the blades spins it faster, which in turn spins the blades in the intake stream faster. When you open the throttle, more air goes into the engine. When more air goes in, it spins the exhaust blades faster, which pressurizes the intake side, which means more air to spin the exhaust blades, which creates more intake pressure... . and the cycle builds on itself. So, instead of the engine sucking in it's air supply it is being "force fed". Now, instead of sucking on a straw you are drinking from a fire hose. The amount of time that elapses as the cycle builds is referred to as "turbo lag".
So, the gauge is telling you how much pressure is inside the intake system compared to the atmospheric pressure. If you are seeing a reading of 15 psi that means that the pressure inside the intake manifold is 15 psi higher than the pressure of the air you are breathing. It's all relative, because the pressure of the atmosphere around you varies constantly depending on weather, temperature and altitude. Just like when you put air in your tires, the pressure gauge is telling you the difference between the pressure inside the tire and outside the tire.
Lastly, a supercharger is basically a turbo charger that is driven mechanically off of the crankshaft, instead of being driven by the exhaust gases like a turbocharger.
The debate about which is better has gone on since the inception of the internal combustion engine. Each has it's strengths and weaknesses and there's no way I'm opening that can of worms. :-laf