It's a physical effect.  Strings, particularly the thicker ones, vibrate at different pitches depending on the size of the vibration.  Strobe tuners allow you to see this, even with some background noise, and there are several strobe tuner apps available for the iPhone.

How to pluck and what part of the strings bending pitch to tune is an artistic musical decision.  You get to decide which part of each strings pitch profile is most important to your music.  If it's the pluck, or attack, and you rarely let notes linger, you want to tune the pluck or earliest pitch.  If you are vocally harmonizing to a long sustain, then you might want to accurately tune that part of the strings "singing", and let the pluck be sharp.  Or something in-between.