I suppose the first question, for many of you, is what is umlaut? This is a term people like me throw around a lot, often without stopping to consider the confusion among non-philologists. “Non-philologists”, I suppose, is another way of saying, 99.999% of the human race. ;)
Put simply , umlaut is a phonological process whereby the pronunciation of a vowel is influenced by the vowel (or semivowel) in the subsequent syllable. This sound change comes in many different flavors, some more common than others. In the Germanic languages, umlaut frequently refers to a more specific sound change where vowels are raised or fronted  under the influence of i or j in the following syllable. For these reasons, when speaking of Germanic umlaut, the synonymous terms “i-mutation” and “fronting” may be encountered (you will sometimes also see “palatal umlaut”). This sound change occurred in all the Germanic languages except for Gothic. 
As for why one vowel changes under the influence of another, there are two basic views. Randolph Quirk and C.L. Wrenn may have summarized it best: “The generally accepted phonetic explanation […] is that the high front i or j palatalised the preceding consonant and that this in turn pulled the vowel of the stem towards its own position, raising or fronting it. […] This theory may be called ‘mechanistic’, because it is based entirely on the assumed workings of the speech-organs. An alternative explanation is that in pronouncing the back vowel in the root-syllable the speaker unconsciously allows his mind and his tongue to ‘anticipate’ the i or j that is to come in the immediately succeeding syllable, […]. This is a ‘mentalistic’ or psychological theory of i-mutation. The orthodox view of articulatory influence through the consonant is a theory of attraction and assimilation, while the mentalistic view is one of anticipation.” 
Since some eyes may be glazing over at that, let me make this a little more plain: i is basically the highest, frontmost vowel there is. It’s so high and so fronted, that it can’t help but pull other vowels toward its point of articulation; not to do so would put a much greater strain on the speech process, and if there’s one sure thing we can say about the speech process, it’s that it’s lazy. It will always take the path of less resistance and least strain on the speech-organs.
Perhaps a few examples would help to make i-mutation clearer. Let’s consider Old English gold “gold”, and observe how the process worked. OE gold was originally *guld (cp. Old Norse gull, and Gothic *gulþ, attested only in the dative singular, gulþa). The suffix used to form the adjective “golden” is still clear in Modern English. We should have expected very early OE *gulden, which mutated by umlaut into gylden “golden”, the u “fronting” into the corresponding short front vowel, y. Subsequently, under the operation of a different sound change, the vowel is the noun, *guld, was lowered, giving us gold. In Modern English, the signs of umlaut in “golden” are long gone, but they were quite clear in OE gylden.
How about another? Think about Modern English “old”, “older”, “oldest”. Do you see where I’m going with this one? In more archaic English, of the type Tolkien often used to represent the speech of Gondor and Rohan, we see the forms, “old”, “elder”, “eldest”. Here umlaut survived into Modern English — for a while. Let’s have a look at the antecedent forms.
In Old English, these were eald, ieldra, ieldest. So, hmm, where’s the i or j we need to account for the fronting of the diphthong ea to ie …? You have to go further back. The original comparative and superlative suffixes in Proto-Germanic were *–izo, *–isto — there’s our vowel! By the time of Primitive West Germanic, the comparative had rhotacized to *–iro , while the superlative remained unchanged. By the time of Primitive Old English, this would have given us first eald, *ealdira, *ealdist, which would in turn have mutated by umlaut into the recorded forms, eald, ieldra, ieldest. These are the early West Saxon forms. In the later “classical” West Saxon of around the year 1000, these had eroded into eald, yldra, yldest. In the Mercian dialect (Tolkien’s favorite and mine), the situation looks at once more familiar: Old Mercian ald, eldra, eldest.
If you speak any German, this should look equally familiar, as the Modern German forms are alt, älter, ältesten. As you can see, in German, the vowel experiencing umlaut is still written as the same original letter, but a diacritical mark is placed over it to indicate the umlaut, and the pronunciation is indeed raised or fronted (from a to e).
As the last example shows, umlaut is still very much with us today. It’s typically associated with German, from which the process takes its name (umlaut is um “after” + laut “sound”). A couple more examples from German: Frau, but Fräulein. And schön from Old High German scóni. But though usually thought of in connection with German, it’s still present in English too.
Here’s something I’ve been building up to. Ever wonder why it’s Anglo-Saxon but English? Or why it’s Anglia, but England? Had I begun with this question, you might have been scratching your heads, but now you know the answer: the change from a to e is umlaut! In Old English, the angle were the Angles (as in Angles, Saxons, and Jutes), but the adjectival form of their ethnonym was englisc (originally *anglisc, acted upon by umlaut). And this is where we come to Tolkien. You may have wondered whether I’d live up to that promise, so dense has been the discussion up to now! Hopefully, you’re all still with me.
Tolkien was far more expert than I in matters of Germanic sound laws. He owned books with impressive titles like Laut und Formenlehre Altgermanischen Dialekte [“Sound and Morphology in the Old German Dialects”], and he read them in their original languages and made annotations and corrections in their margins. He was better versed in umlaut than I will ever be and would surely have found plenty to niggle at in my explanations of it. It should come as no surprise, therefore, that he worked examples of umlaut into his fiction (as he did so many other philological elements). I can think of three instances in The Lord of the Rings. If I’ve successfully communicated the basic idea behind umlaut, can you think of any? Pause here to put on your thinking cäp.
Here’s a hint: just like Angle/English, the examples of The Lord of the Rings are topo/ethnonymic.
So. Just as we have Angle, but English, Tolkien gives us Dunland, but Dunlending and Dunlendish, where the i in the final syllable causes the a in the second to be raised to e. This is as straightforward a case of umlaut as you could wish for. Interestingly, the words dún-land “down, hilly land” and dún-lendisc “hilly, mountainous” are attested in Old English, as are uppe-land and up-lendisc, both pairs clearly demonstrating umlaut in the real world. Returning to Middle-earth, another example from Tolkien follows the same pattern: Sunlands, but Sunlending, each used only once in the novel, in reference to the far southern regions of Harad.
And finally, again in connection to Harad and the Sunlands, what about the curious Shire word, Swertings? “Swertings we call ’em in our tales; and they ride on oliphaunts, ’tis said, when they fight,” Sam tells Gollum . This is a little less obvious, but it must be the umlauted form of *Swartings, derived from swarthy, a word Tolkien often uses of the Harad-folk. The word swart or swarthy comes from Old English sweart “black” (Mercian *swart), cp. Old Norse svartr “black” and Modern German schwarz. In Old Norse, there is a proper name Svertingr, which probably carries a swarthy meaning and shows umlaut from svartr; likewise, probably, for the Swerting in Beowulf, though we can’t really say much about him. Also in Old English, swertling was used to gloss the Latin ficedula, a small passerine bird, dun or drab (or swarthy) in color. Today, ficedulae are Old World flycatchers of the order Passeriformes, but Bosworth/Toller supposed that swertling might be the titlark, a bird of the same taxonomic order, but different in family, genus, and species.
There you have it. Both real-world and Middle-earth examples, side by side. Put there, in fact, by one of the most gifted Germanic philologists the world has ever seen. Should we be surprised? Of course not! Is it interesting? Well, I certainly think it is, and I hope you agree. :)
More, and lengthier, notes than usual …
 As complicated as this must sound to a lay reader, believe me, I have simplified it. The whole process is made more difficult by the fact that the i or j (especially the latter) frequently disappeared by the time the words in question were being set to parchment. Other processes of sound change might subsequently alter the vowels of the stem, inflexions, or both. Inflexions may have been lost entirely. Exceptions may have preserved root vowels where we would have seen umlaut. And so on. But at its simplest: i-mutation is the raising or fronting of a root vowel under the influence of i or j in the following syllable.
 Fronting and raising aren’t the same thing, though they’re closely related. Each vowel, like all speech sounds, is articulated at a certain location somewhere in the speech cavity, somewhere from the lips to the glottis (front to back), from the soft palate to the lower jaw (top to bottom). Fronting means that the articulation of a vowel moves from the back of the speech cavity toward the front (e.g., fool to foot to fur); while raising means a vowel moves from the bottom toward the top of the speech cavity (e.g., frond to friend to frill). Try pronouncing these groups of words and pay attention to how your tongue moves inside your mouth: forward with the first group of words, then upward with the second group.
 Actually, it might have occurred in Gothic, but two problems: the vast bulk of the Gothic we have is from the 4th century, which predates the umlaut process; and if umlaut did occur in Gothic, we don’t have any later texts that would show evidence of it. One would think it should have occurred in Gothic, and this has occasionally been alleged by scholars, but we have no clear evidence of it in the surviving corpus. By the way, i-mutation isn’t an exclusively Germanic process — there are examples in the Romance languages as well — but it was much, much more prevalent in the Germanic language family than in others.
 Quirk, Randolph and C.L. Wrenn. An Old English Grammar. 2nd ed. London: Methuen, 1957, pp. 153–4.
 Rhotacism is another sound change in the Germanic family, whereby Proto-Germanic z became a rhotic, or r-like sound. Like i-mutation, this occurred in all the Germanic languages except Gothic (er, maybe; see note 3). Example: PG *deuzom gave Gothic *dius (attested in dative plural diuzam), preserving the z; but this was rhotacized throughout the rest of the family: Old Norse dýr, Old Frisian diar, Old Saxon dior, Old High German tior, Old English déor “wild animal (> deer)”. Why should the z and r sounds be related? Ask Antonín Dvořák!
 In draft, it was Gollum, not Sam, who called the Haradrim Swertings.