In a position feedback PID controller the integrator component is the sum of all the position error. You could call it the hysteresis, or history of error. Regardless, it is like a running average error and tells your control system how consistently off the mark it is in general. Typically the integrator gain is small to keep a stable system, so that you don't have a strong P or D response to the I induced response itself (the gains reacting to each other's reactions is a source of instability).
Before you crank up the integrator gain, you should ideally implement an open loop control set point and use it with a P, then a PI controller, and PID only if necessary. This is called 'feed forward' controls. In feed forward, you essentially run open look with a dynamic P response and often a slow I response to ensure you adjust dynamically for temperature and other dynamic variations that would effect long term error. D response is only necessary if you have to prevent overshoot while needing very high P gain.
If you have a feed forward system and you see high integral buildup for certain positions and none for others, it means your feed forward is not tuned properly to account for the known variables.
The most layman way of thinking that occurs to me is if you are throwing a ball back and forth, but you always miss to the left of your target, you adjust to the right until you get it on target every time. That's integral gain working in your much more complicated brain, which had a feed forward system to determine where and how to throw the ball initially.