I-MUTATION (also known as "i-umlaut") is the raising and fronting of a root vowel in anticipation of "i" or "y" sound in a suffix.
This sounds complicated, but it's really pretty basic. You probably do it every time you speak.
Think of the difference between the -o sound in the do of "How do you do?" and that of the last word in "How are you doing?" The last word of that sentence might be written *diwin if it were spelled phonetically the way the average modern American pronounces it. When that -o- shifts up to an -i-, that's i-mutation.
Say the vowels slowly, in order, out loud, shifting from one to the other without pause -- "A-E-I-O-U" -- and feel how they "happen" in different parts of your mouth. Philologists talk of "high" and "low," "front" and back." "A" is the lowest, "backest" of our vowels, and that's why doctors tell you to make that sound when they want to look down your open throat.
I-mutation is caused by the very human habit of laziness: taking the shortest distance between two points. The plural of man in ancient West Germanic, the ancestor of Old English, used to be a word something like *manniz. The speakers "cheated" on the first vowel in the word to be in position for the second vowel. It's the same thing you do with doing. It doesn't change the meaning of the word to do so.
So after hundreds of years of this, the plural came out as *menniz, or something similar, when people said it. Eventually, the shifted vowel itself comes to stand for the plural, and since laziness dislikes doing the same job twice, the syllable at the end of the word slowly shriveled and dropped off.
Most such suffix vowels were gone by the Old English period, but their effects remained and in a few cases still do. Some of the main places you can still find evidence of i-mutation are: