Two dots, a different vowel
German has three vowels that English has no letters for: ä, ö and ü. The two dots are not an accent mark and not decoration. They make the vowel a different letter, with a different sound, and often a different word.
Swap an Umlaut for its plain partner and you have said a different word: schön (beautiful) becomes schon (already), Mütter (mothers) becomes Mutter (mother).
Hear the difference
schon
already
schön
beautiful
Mutter
mother
Mütter
mothers
Vater
father
Väter
fathers
ä: the one you can already say
ä is the vowel of English bed and men, long or short. Do not turn it into the ay of day: Käse is not kay-se. German vowels stay steady and do not glide.
Words with ä
Bär
bear
Käse
cheese
spät
late
The dots also appear in äu. It sounds exactly like eu, close to the oy of boy: Häuser (houses), Bäume (trees).
ö and ü: the two you have to build
Both are rounded front vowels. Say the English vowel, hold it, and then push your lips forward into a small circle without moving your tongue. The tongue stays where it was; only the lips change.
Words with ö
schön
beautiful, nice
hören
to hear
möchte
would like
Words with ü
Tür
door
fünf
five
müde
tired
In everyday sentences
Listen and repeat
Ich möchte einen Kaffee, bitte.
I'd like a coffee, please.
Ich bin heute sehr müde.
I'm very tired today.
Die Tür ist offen.
The door is open.
Kannst du mich hören?
Can you hear me?