If you really want to understand how a PID is implemented, I highly recommend you to read the following blog series:
They are short and well explained, also is the implementation used in the pidtuner tool.
Regarding the PID parameters, some PID implementations use PID gains directly (so called parallel or ideal form), other use times (so called standard form). That is why the tool provides both, then the user must use the ones that their PID requires. Read:
Hope this is helpful.