In Modern German, back and open vowels alternate with raised and fronted pairs in morphologically related words.

In terms of spelling, the "umlauted" vowel is usually marked with a diaresis (two dots above the vowel, often just called an umlaut):

<ü> in der Schüler vs. <u> in die Schule
<ü> in die Mütter vs. <u> in die Mutter
<ö> in lösen vs. <o> in los
<ö> in der Mörder vs. <o> in der Mord
<ä> in das Hänchen vs. <a> in der Hahn
<ä> in die Männer vs. <a> in der Mann
<äu> in die Häuser vs. <au> in das Haus

In terms of pronunciation, the "umlauted" vowel is fronted and (in the case of [a] and [a:]) also raised with respect to its non-umlauted pair:

Some examples

NHG exploits umlaut to mark contrasts, sometimes with and sometimes without another marker.
Plural vs. Singular Noun Phrases die Männer vs. der Mann
die Hände vs. die Hand
die Brüder vs. der Bruder
die Mütter vs. die Mutter
die Häuser vs. das Haus
Comparative and Superlative vs. Positive Adjectives klüger vs. klug
dümmer vs. dumm
gröber vs. grob
öfter vs. oft
näher vs. nah
stärker vs. stark
Subjunctive (Konjunktiv) vs. Indicative forms of verbs er führe vs er fuhr
er böte vs. er bot
er bäte vs. er bat
Present Singular 2 and 3 vs. Other Present Indicative forms of verbs with a or au in their present stem er schläft vs ich schlafe
er läuft vs. ich laufe
er lässt vs. ich lasse
Word Formation relationships das Bäuchlein vs. der Bauch
das Büchlein vs. das Buch
die Größe vs. groß
die Stärke vs. stark
die Länge vs. lang
ärztlich vs. der Arzt
tödlich vs. der Tod

For interest

The ultimate origin of umlaut is an original high front vowel in the following syllable, to which the vowel of the preceding syllable was assimilated, and this explains why open and back vowels are raised and/or fronted.

If we trace the forms back though the history of German, as the following table shows, the origin of the change appears to be in the Old High German period and before. Note that we must reckon with other sound changes, such as the NHG Diphthongization, e.g. <hiuser> becomes <Häuser>, vowel lengthening and the change of [u] to [o] before [n].

Modern German Middle High German (1050-1350) Old High German (750-1050) pre-Old High German (before 750)
<Häuser> "houses" <hiuser> <husir> *[hu:siz]
<Würfel> "dice" <würfel> <wurfil> *[wurfilaz]
<Söhne> "sons" <süne> <suni> *[suni:z]
<höher> "higher" <hoeher> <hôhiro> *[ho:xizo:]
<Gäste> "guests" <geste> <gesti> *[gasti:]

The traditional explanation (still taught in history of the language courses) is that we have a primary umlaut of (pre-OHG or West Germanic) short */a/ to short /e/, which explains Old High German <gesti>, and then we have a secondary umlaut between the Old High German and Middle High German periods, which explains all the other umlauted vowels. It also seems to fit in nicely with the idea that the umlaut products were allophonic variants of the non-umlauted vowels, which became phonemicized only when the conditioning factor (the <i> in the following syllable) disappeared due to the loss of distinctions between unstressed vowels which we observe from the late OHG period onwards. The idea of a secondary umlaut also explains why there are two umlaut products for short [a]: first <gesti>, MHG <geste>, from pre-OHG */gasti:/, and second <mahti> MHG <mähte>, from pre-OHG */maxti:/.

However, there are some intractable problems with this explanation.

To begin with, there are data which do not fit within the language. Take the verbs with Rückumlaut, which have umlaut in the present but no umlaut in the preterite. Examples are MHG <hoeren> and preterite <hôrte>, MHG <füllen> and preterite <fulte>, and MHG <nennen> and preterite <nante>. (In Modern German, the Rückumlaut verbs have been regularized by analogical levelling, except the group senden - sandte, nennen - nannte, etc.). In Old High German, these verbs would have been written <hôren> and preterite <hôrta>, <fullen> and preterite <fulta>, and <nemnen> and preterite <namnta>, pronounced, if the primary and secondary umlaut view is correct, [ho:ren] - [ho:rta], [fullen] - [fulta] and [nemnen] - [namnta]. This means that we must assume that some Rückumlaut verbs (except those with short [a]) had lost their umlaut in the present in Old High German and regained it by Middle High German. Some scholars even assumed an "underlying" [i] to explain this unique phenomenon of a sound change with no apparent cause.

Secondly, taking data from closely related languages, we can see that umlaut affected them as well. In the case of OHG <fullen> and preterite <fulta>, for example, we have Old English <fyllen> "to fill" (and preterite <fylde> by early analogical levelling), so, if the primary and secondary umlaut view is correct, we must assume that umlaut occurred independently in the two languages. The same happened for OE <cyssan> (pret. <cyste>) and OHG <kussen> (pret. <kusta>), which gives MHG <küssen>, pret. <kuste>, OE hîeran "to hear" (<îe> is the umlaut of OE <êa>), and so on.

It would be simpler all round to assume that umlaut took place in prehistoric times and to explain the reflexes in the individual languages on this basis. However, this explanation raises a further series of problems.

The first is: why did the scribes indicate one umlaut product, <e> from short */a/, and not the others?

The answer is that the scribes wrote German within an existing tradition of Latin writing. Latin has five vowel symbols <a>, <e>, <i>, <o>, and <u>, and the <e> readily served as the symbol for the umlaut of <a>. So a river name such as <BRANCIA> is easily rendered as (the river) <Brenze> (Berger 1999:71-2), because umlaut would have taken place. But just because the Latin name of modern Zürich, <TVRICVM>, is rendered <Zurih>, does not mean that this word would have been pronounced without umlaut - it would still have been pronounced /tsy:rix/. The pronunciation would have changed, but the traditional Latin spelling of names did not, just as in English the spelling <thought> reflects the original fricative pronunciation which we still find in Scots (and also in German <dachte>!). After all, the scribes knew how their language was pronounced, and Latin orthography provided a framework for the spelling even though the pronunciation had changed. We can therefore safely assume that OHG <husir> was pronounced [hy:sir], <hôhirô> was pronounced [hø:xiro:], and so on. It just took the spelling a while to catch up with the changes in pronunciation. Interestingly enough, umlaut is not regularly marked until the Modern German period: the standardized grammars of Middle High German give a false picture of uniformity.

If that is so, and this is the second problem, why do some instances of pre-OHG short */a/ appear to be umlauted (e.g. <gesti> "guests"), while others do not (e.g. <nahti> "nights")?

The answer to this one is that we must assume a rather more complex change.