I'm attempting to get a better handle on how the ADC works exactly. How would the ADC operate in comparison from A to B

A) using a 5vdc board with 7.4 to 8.5vdc as a power supply and being regulated to 5vdc
B) using a 5vdc board with 3.7 to 4.2 vdc supplied after the regulator (no voltage regulator)

if one were to use the formula to convert 1023 to 5vdc, would 613 be 3vdc? or is the vcc voltage a refference point to the 1023?

The ADC compares the input voltage to the voltage reference, for which there are several possibilities. Default is Vcc, whatever that happens to be.

To get the correct voltage from an ADC reading performed by your program, you must use the correct value for the voltage reference.

B) using a 5vdc board with 3.7 to 4.2 vdc supplied after the regulator (no voltage regulator)

What?

Of course, the reference should be stable and it should be known.

A separate regulator/reference tends to be more stable. If you are powering a bunch of "stuff" from the Arduino's built-in regulator the voltage can vary when the load varies. (It's regulated but nothing is perfectly regulated.

if one were to use the formula to convert 1023 to 5vdc, would 613 be 3vdc? or is the vcc voltage a refference point to the 1023?

It's a ratio… ADC/1023 = Vin/Vreff. So yes. 613/1023 = 3/5 (with integer rounding).

Thank you, that’s what I figured but in my quick google search I didn’t find a definite answer

A minor point: the correct divisor is 1024, not 1023. This has been debated many times.

If the voltage reference is 5.000V, then “1023” means only that the voltage is 4.995V or higher.

Many sensors are ratiometric (they output a ratio of their supply).
A pot, a thermistor, ADS712 current sensor, pressure sensors, etc..

A pot is a good example.
If you turn it half-way, it outputs half of it's supply voltage (assuming linear pot).
If VCC is 5volt, then the pot outputs 2.5volt.
If VCC is 3.3volt, it outputs 1.65volt.

Much the same with the A/D, but opposite.
2.5volt on an A/D input with 5volt Aref (supply) outputs a value of ~512 (assuming 10-bit A/D).
1.65volt on an A/D input with 3.3volt Aref also outputs ~512.

So for a ratiometric sensor, absolute supply voltage doesn't matter.
Leo..

LandonW:
I'm attempting to get a better handle on how the ADC works exactly.

Have you read the relevant section of the datasheet?