From Wikipedia, the diaeresis …
… is used to indicate that the e is to be pronounced separately from the preceding vowel (e.g. in the word “reëntry”, the feminine name “Chloë” or in the masculine name “Raphaël”), or at all – like in the name of the Brontë sisters, where without diaeresis the final e would be mute.
I’ve worked to set up my writing environment to prefer this ornate style of writing.
Anecdotally, my nieces struggle with the “change of the rules”, and so they asked me to explain it to them. I possess no recollection nor any idea how I learned the difference.
By the way, on a Mac one can get a diaeresis via ⌥u and then e, or o, or a, or u, or i, or even y.