Powering a PC monitor with batteries
24 W for 8 hours is 8 x 24 = 200 Wh (watt-hours). (Don't worry about the calculation error. It is nothing compared with the ill-defined battery specification.)

Since \$ P = VI \$ we can calculate the amp-hours required for a particular battery voltage. With a 12 V battery \$ Ah = \frac {P}{V} = \frac {200}{12} = 17 \ \text {Ah} \$.

Ah is a popular measurement but it is only useful when comparing systems with the same voltage as, for example, in auto batteries. Wh is a true measure of the energy stored in the battery and allows much simpler comparison.



From the comments:


  First I have a monitor that has uses 12 V and a micro controller that uses 5 V. I am assuming since these are in parallel they will equate to 17 V draw on the source. 


No. Imagine you have two 12 V lamps in your car. Does that "equate to a 24 V draw" on the 12 V battery? Obviously not. Parallel connection increases current, not the voltage.


  The panel is drawing ~24 W and the controller 4.5 - 5.2 W under load (~32 W max for system). If I made cells consisting of 5 - 4.2 V 18650 batteries in series equaling 21 V cell, would this give me 256 Whr / 21 V = 12.2 Ah?



You need a 12 V supply for the monitor.
You need a 12 V to 5 V converter for the controller.
The monitor and converter are wired in parallel.
You need 200 Wh for the monitor.
You need 5 W x 8h = 40 Wh for the controller.


Where you use a voltage converter you need to include a factor for efficiency. If the 12 V to 5 V converter has an efficiency of 80% then the input power required = \$ \frac{P}{eff} = \frac {5}{0.8} = 6.25 \ \text A \$.