Usually when you sense AC voltage, you get a sinusoidal waveform. However, for AC voltage sources, Arduino analogRead ( ) only presents positive voltage values. So what's basically happening? Is it skipping half the cycle where the voltage is negative. If so, how can I account for that when I try to use the voltage values to calculate for total electrical energy generated by the source (piezoelectric film) and avg. power?
I guess that you are showing the voltages there directly calculated from
analogRead() and that you connected the AC voltage directly to the analog input (and it's ground with the Arduino's ground).
The reference for every communication/voltage measurement/digital input is ground. Naturally the Arduino can only measure voltages inside of it's own supplied voltage range, meaning 0 to 5V for the Uno. If you apply a negative voltage (relative to ground), you will measure a voltage of 0V. You can even damage the microcontroller, if the negative voltage is too high.
If you have negative and positive voltages, you should offset the signal at the analog input pin, so that the resulting signal stays in the range from 0 to 5V. You can do this with a simple voltage divider. Use two resistors with the same value (for example 5kOhm), so that you will get a new zero point at 2.5V. The AC signal will then variate around that value.
For what is happening inside the microcontroller:
The ADC is responsible for measuring the voltage. For this the voltage on the analog input pin will first be sampled (loaded onto an internal condensator) and then step by step compared to the voltages, that the ADC generates by dividing the supply voltage range into 1024 distinct values (10 bit). Again the reference, to which the voltage is measured/compared, is ground. It cannot generate negative voltages, thus cannot compare them.