Since we are talking layman’s terms for me, now that we know what P does, what about I and D?
With an Auber PID the "I" will help to control initial overshoot and I believe it also helps tighten the oscillation up/down around your set temp. Chop has this covered in a post above.
Basically every PID device can be different but Auber is pretty consistent with their PID settings and temp control algorithm (steps to solve the problem) using the PID settings.
A little bit more layman's explaination here.
P - I believe with an Auber controller the "P" value will go full power to heat until you get within the P-settings number of degrees below the set temp. So if P=7, you get full power until 7 degrees from the set temp then it cuts power down as it approaches set temp and when it hits set temp or is over P will not provide any power. think of "P" as like brute force come up to temp power setting provided you tuned "P" properly.
I - starts to kick in when the controller reads that the smoker temp is around the set temp. If it is below the set temp and moving lower "I" will feed more power to get to set temp. If above set temp and moving higher "I" will cut power to get to set temp. It does this to dial in and hold close to the set temp since "P" is often no longer in the mix. Also when set properly with "P" as "P" drops off "I" value will be used in calculation to help pick up to hit set temp OR help fight against "P" so that you don't way overshoot your set temp.
Think of "I" as the cruise control to keep you on target with your set temp provided you tuned "I properly."
D - is kind of the magical intervention fairy godmother looking at temp rate change. It frequently looks back in time to see what the temp was a few seconds ago and it compares to current temp and reviews your set temp. If you the rate of change is too drastic in any direction "D" will override "P" and or "I" to get or keep you around the set temp. So when you open the smoker door the temp drops like a rock and "D" comes to the rescue. Or if temps sky rocket to fast "D" attempts to come to the rescue.
Think of "D" as setting that handles uncommon temp change situations
With different PID manufacturers they do may do completely different algorithms (steps to solve the problem) therefore their PID numbers may be widely different.
The Auber behaves this way and is explained as so in their manual... provided I remembered it all correctly so you or anyone else please feel free to correct anything I may have misstated hahaha.
I hope this info helps :)