For a while, I’ve been interested in a subset of English nouns that are formed by suffixing /-th/* to an adjective. What’s peculiar about them is the alternation of vowel sounds between the lexical categories: the nominal form has one vowel, and the adjectival has another. Here are two common examples:
long - length
strong - strength
The chart below visually demonstrates the process. It’s essentially a diagram of your tongue’s location in your mouth during the execution of vowels. For the /ɑ/ in long and strong and your tongue is low and retracted, and for /i/ of feet its pushed forward and up. A quick test on your own will confirm this.
Figure 1: /ɑ/→/ε/
During I-Mutation, the sound /ɑ/ is pulled upward and forward, and settles at an intermediate between /ɑ/ and /i/, namely /ε/.
This process doesn’t happen spontaneously: it’s environmentally conditioned, meaning there must have been an /i/ nearby to trigger the change. Historically, the /-th/ suffix was more like /-ith/ or /-ithu/, so it actually contained the sound /i/ (”ee”). So, in anticipation of the vowel in the suffix, speakers would move the stem vowel incrementally closer to /i/, and they stopped off somewhere in between. Later the /i/ was deleted in a process called syncope.
Some linguists have attributed I-Mutation and similar processes to human laziness, claiming that speakers try to expend minimal effort to pronounce the vowels of a word. However I think it’s more likely a constraint on articulation: the tongue is physical object, and must actually move from one position to another. Over time, this transition becomes audible, and begins to color the vowels. Eventually, the “slide” between sounds becomes part of the accent of that language and becomes established (think of a Spanish speaker trying to say home, they have much trouble mastering the slide from “oh” to “oo” packed into that vowel). But I digress.
For the other words in this class, an original I-Mutation has been obscured by other processes. For example, in the following words, the adjective was derived from the noun and was preserved. Later, the Great Vowel Shift changed the sound of the noun, but left the adjective intact:
fūl → filth  ‡
Figure 2: /u:/→/ɪ/
Later, fūl (pronounced “fool”) became foul by means of the Great Vowel shift: /u:/→/au/
hāl → health /ɑ/→/ε/ (See Fig. 1) 
Here, the Great Vowel Shift covered I-Mutation’s tracks by changing hāl (pronounced “hall”) to whole and a less commonly used word hale. The likely explanation for the two results is that there were two competing pronunciations of hāl, and each went a different way with the Great Vowel Shift (and it probably has to do with the influence of /l/ on the preceding vowel … don’t ask).
In other instances, it’s more complicated. The Old English word slaw had the same vowel of modern (cole)slaw, but the /w/ was actually pronounced /slɑw/. I-Mutation pulled the /ɑ/ forward:
Figure 3: /ɑ/→/æ/
… and we were left with slæwth (rhymes with mouth, which stuck around for quite a while in Old English. However, in Middle English, slaw changed roughly to the modern pronunciation slow, and speakers – conscious of the relationship between slow and slæwth – futzed the vowels.  The important thing is that the /th/ stuck around.
One final example is young - youth, although the story of /ng/ is a bit convoluted.
All of the derivations we’ve examined thus far were formed in Old English and were fossilized. Since then, the /-ith(u)/ suffix ceased being a preferred way of forming nouns, and has been abandoned in favor of suffixes like /-ness/. Linguists say suffixes like /-ith/ are no longer “productive,” because they aren’t actively used in the generation of new words, despite remaining as vestiges in certain words and phrases.
Although /-ith/ suffixation fell into disfavor, some words have nevertheless been formed to superficially resemble the fossilized forms through the process of analogy. For example, wealth was formed from well under analogy to health . Here, both forms have the same vowel. This is probably evidence that analogy was afoot, because speakers don’t have conscious access to the rules of phonological rules like I-Mutation: they just added /th/ and called it a day.
On the other hand, some analogized words do exhibit change in the stem vowel, just not in the manner we know to be consistent with I-Mutation. Depth was formed under analogy with length , but this vowel alternation is inconsistent with I-Mutation. Similarly, analogy to length is also the provenance of width (from wide).
One final example is the word breadth, derived from broad under analogy with length.  This example does obey the rules of I-Mutation, but we can tell from its first attestation (1520) that I-Mutation had already swept through the Old English lexicon by the time it was coined. If it wasn’t one of the originally I-Mutated forms, it’s probably just a faithful imitation of the long – length alternation.
Most interesting is the emergent tendency to analogize to length for words relating to dimensions. It’s as if speakers wanted wide, deep, and broad to behave like long so that they could all be one neat set. This “ironing out” the kinks in the language is still happening today (although there’s some out there who would fight it). The noun height is traditionally formed from the adjective high, but you can often hear people saying heighth! It’s not hard to imagine a day when heighth has become accepted, and all the words for dimensions end in /th/.
* This sound is usually transcribed with a theta /θ/, but here I use /th/ for typographical simplicity.
† Technically, it was probably /ɔ/→/œ/ which was subsequently unrounded to /ε/. That’s why we have “o” in spelling today, but you get the idea.
‡ This almost certainly involved an intermediate /ʏ:/ which was later unrounded to /ɪ/.
Mitchell & Robinson. A Guide to Old English. Blackwell Publishing, 2001.