Electronic – the importance of peak factor and form factor

If in fact you mean crest factor then the wiki article linked provides a good source. So, for instance, crest factor is a useful term when applied to multimeters: -

Crest factor is an important parameter to understand when trying to take accurate measurements of low frequency signals. For example, given a certain digital multimeter with an AC accuracy of 0.03% (always specified for sine waves) with an additional error of 0.2% for crest factors between 1.414 and 5, then the total error for measuring a triangular wave (crest factor = 1.73) is 0.03% + 0.2% = 0.23%.

For other applications, consider an audio amplifier designed to handle an RMS signal of 1 volt at the input. If we didn't understand anything about the crest factor of the actual signal into the amplifier, we might design it to work with 1 volts RMS sinewaves and neglect the fact that for an audio signal, the crest factor is quite high. Neglecting the crest factor would mean that the amplifier would distort heavily on the audio peaks: -

Is there any explanation or usage that shows importance to why the quantities for those ratios are selected and placed that way?

For simple waveform shapes like sine, square, triangle and sawtooth, the crest factor is unimportant to state because, mathematically the signal shape defines what the peak to RMS value actually is.

As for form factor the Wikipedia article explains it sufficiently but I'll focus on one particular point of relevance: -

Digital AC measuring instruments are often built with specific waveforms in mind. For example, many digital AC multimeters are specifically scaled to display the RMS value of a sine wave. Since the RMS calculation can be difficult to achieve digitally, the absolute average is calculated instead and the result multiplied by the form factor of a sinusoid. This method will give less accurate readings for waveforms other than a sinewave.

So, it comes into play when cheaper voltmeters are used to measure (or infer) the RMS value of a signal. We are nearly always interested in the RMS value and more expensive equipment will have "true RMS" measurement capabilities but cheaper equipment will use an inexpensive signal rectifier and then low pass filter that rectified output to make an estimate of RMS. If the signal is a sinewave then, the meter knows that it has to multiply the averaged-rectified value by 1.1107 (\$\frac{\pi}{2\sqrt2}\$) to predict the RMS value. Of course, it doesn't use a multiplier; it uses a little signal gain on the average signal before feeding it into its ADC.

But, the problem comes when the signal is not a sinewave. If it were a square wave then the predicted RMS value would be 11.07% too high because the from factor for a square wave is unity. Likewise, if the signal were a triangle wave (form factor \$\frac{2}{sqrt3} \approx 1.1574\$), then the signal RMS would be predicted to be about 4% low.