Difference between online PID and offline PID tuning?
The article in Wikipedia is describing the method of "tuning" the PID parameters to the "process" it is trying to control. It suggests, that if you can run the process "offline" i.e. open-loop, you can measure the open-loop response of the process (to a step stimulus) and use knowledge gained to be able to predict the PID parameters most suited to improve/perfect the process.

Some processes cannot be taken off-line (hence they are called "on-line") and different methods are suggested for determining the optimum PID parameters.