Well, I'll risk oversimplifying to help. I expect someone with a bit more experience will correct me or add.
Gain is the servo reaction time. When too high the servo is unstable and will react to noise readily, when turned down it doesn't react to velocity changes quickly enough and becomes sluggish, often causing finish problems. This is the P in PID.
I is the stiffness. This controls how much movement the motor makes before countering that movement with a reaction. When this is too high it will sound like the motor is cogging, brrrr. When too low the axis will be sloppy and may not hold tolerance. You can think of this as the holding torque of a non-moving axis. This is the I in PID.
The last parameter is the undershoot/overshoot, or the tolerance to velocity changes. Not all drives will give control over this parameter. It contributes to the ability for the axis to hit a repeatable value every time. If the value is too low, the axis will overshoot the target point and may not repeat very well. When the value is too high, the axis may undershoot the target point because it prematurely halts axis movement; reacts too quickly. This is the D in PID.
The fastest part of the loop is usually the gain, then the I, then the undershoot/overshoot. Often the stifness is set lower than the others, with the u/o set high to ensure accuracy. The gain is adjusted until you get a hum, then back down. On digital drives you adjust until the drive spazes out at the slightest velocity command, then back it down.
On DC drive, you can readily tune them quickly with just your ear. Wait for the hum, dial back 1 to 1.5 steps. For I or Current, adjust it up until you get a rough sound from the motor and back it down until it smooths out. For u/o you need to use a precision dial indicator or take test cuts. Adjust it up until the finish stops improving, then back down to the last setting.