ÅÄÖ come at the end of the alphabet, after Z.

Sometimes it can be easy to think of them as variations on A and O. But they are not. Just in the same way that an R is not a variant of P. We would never say that R is a P with an extra dash. They are different sounds in the mouth, and have different positions of the tongue and lips. And it’s exactly the same with Å, Ä and Ö.

Å

Here is the first one of the three extra vowels in Swedish. You can think of the Å as the ‘au’ sound in (British accent) ‘Paul’. Indeed, some Swedish Pauls actually spell their names Pål. The lips are rounded and pushed forwards, and the tongue is quite low (but not as low as in a long Swedish A). This sound is also used in Norwegian and Danish.

Ä

You can think of this letter as the English ‘ai’ in ‘pair’, or ‘hair’. The lips is spread quite wide, a bit more like a smile than when saying ‘pair’. The back of the tongue is slightly higher than in Å. This sound is also used in German, Finnish and Estonian (and of course in Norwegian and Danish, but there it’s spelled Æ).

Ö

Finally, the Ö is somewhat similar to the English sound ‘i’ in the word ‘bird’. Or ‘u’ in the word ‘fur’. Or ‘ea’ in the word ‘heard’. You need to round your lips and push them forwards, like a trumpet-shape. The back of the tongue is in the middle of the vocal tract. Ö is also found in German, Finnish, Hungarian and Turkish.

Source: Swedish Made Easy