I was recently asked by a family member why the word naïve has two dots above the i, even though they can only ever remember seeing it written as naive. They thought it was an umlaut, which is added to certain vowels in German (usually ö and ü) to change their pronunciation. A German umlaut (or a “trema” when applied to Dutch) implies an “e” sound, and words can be written with or without the diacritic: e.g. the German for “spoon” can be written löffel or loeffel.
However, the ï in naïve is not an umlaut – it’s a diaeresis, also known as a hiatus. An umlaut signifies a compound letter, whereas a diaeresis signifies that a vowel should be taken as a separate sound from the preceding one – that is to say, that the second vowel is not part of a diphthong. “Naïve” is not pronounced the same as “knave”, but more like “ny-eev”; just as the name Zoë (sadly only rarely seen spelled with the diaeresis nowadays) is not pronounced “zo”, but “zo-ee”.
If you read any old books, you may come across diaereses in words that have, over time, lost them. Words like cooperate were originally spelled with the diaeresis: coöperate, reestablish would be rendered as reëstablish, and seer would be written seeër.
Interestingly, both of the terms for describing this linguistic feature (diaeresis and hiatus) have, in themselves, one of their own: diaeresis and hiatus. Perhaps they should be written diaëresis and hiätus!
The diaeresis is a dying diacritic in English – certain publications like the the New Yorker still retain them as part of their house style, but they are rarely seen or used, besides by grammarians.