Guitar Pickup Harmonics:

Starting with the basics, guitar pickup harmonics (for electric guitars), relate to how a guitar string vibrates and how pickups sense those vibrations. Simplistically, the primary frequency that a string vibrates at is called its fundamental. So the treble E string on a normal 6 string electric guitar has a fundamental frequency of approximately 330Hz. But if you play the same string at the 12th fret you will be playing one octave higher. An increase of an octave is a doubling of frequency, so 2x 330Hz (approx 660Hz). 

But let's go back to that open treble E string. Its fundamental frequency may be 330Hz, but the same string is also simultaneously vibrating at other 'harmonic' frequencies, simple multiples of the fundamental.
So the second harmonic is twice the fundamental frequency (approximately 660Hz, the same as fretting the same string at the 12th fret). There is also a third harmonic (approximately 990Hz ) and so on. The 'harmonic tuning' method many guitarists use regularly exploits the higher harmonics up at the fifth frets.
The links at the bottom of the blog give some great visual representations of these harmonics in action. 

What becomes clearly when you see this diagrammatically, is that the physical position of pickups along the strings length is key to its tonal output. A pickup position immediately under a stationary 'node' on a string will result in the pickup not sensing that node and thus not that harmonic at all.
And of course, these fundamental and harmonic frequencies also vary at every string fretting up and down the fretboard. The pickup design, but also the relative positions of the pickups along the string significantly impact the characteristic of a guitars tone.
And the point along a string it is plucked will also affect the mix of harmonics generated and thus 'sensed'. Hence playing a chord at the bridge generates a more cutting tone than the same chord strummed with a plectrum over the fretboard itself which gives a mellower 'dominant fundamental' tone.

For self builders, the topic of pickup positioning is clearly yet another area for explanation. For anyone with a guitar with fixed pickup positioning, this has hopefully at least given a glimpse as to why some guitars and their pickups sound as they do (and how plucking them differently can drive new tone).
And checkout Ironstone's own page on basic pickup design;
Electric Guitar Pickups Explained via the Full Details link

Related web resources;