The circuit is completely conventional, and is based on the application notes from National Semiconductor. The circuit is shown picture below and as you can see it uses a single IC and a few discrete components. DC to the LEDs is almost unfiltered - C1 is included to make sure the IC does not oscillate, and is not a filter cap. This allows a higher LED current with lower dissipation than would be the case if the DC were fully smoothed, and full smoothing would also require a much larger capacitor. This increases the size and cost of the project - especially important if it is to be used in larger numbers as may be the case with a mixer or analyser.
This could not be simpler. At the maximum level you wish to operate the equipment (as shown on an audio millivoltmeter or oscilloscope with signal applied), adjust VR1 so that the signal illuminates all the green LEDs (L1 is the most sensitive, and L10 indicates maximum level, so L1 to L8 should be lit). If the input is directly from a speaker output, an additional series resistor should be used at the "Aud" input terminal to reduce the level. This can be determined by calculation (I leave this to you) or by experiment. As a guide, for a 50W amplifier, the external resistance should be about 47k ohms.
If you are calibrating the meter for a power amplifier, set the output to a level just below clipping. Adjust the level control until all LEDs are illuminated. This way, if the last LED (L10) lights when you are listening to music, you will know that you are very close to clipping, and the volume should be reduced.