Diaeresis

Silent E
Diphthong
Diacritic marks
Grave accent
(diaeresis is also a poetic break in verse)

Use?

It is used to show that the adjacent vowel to another vowel is pronounced separately.
Usually, in English, we pronounce two Vowel written right next to each other as Diphthong.
We use this mark (as in the one above e of Noël) to show that they are pronounced separately.
coöperate
reëlect
 The New Yorker is one of the rare publications that still use the diaeresis sign.

Should we use it?

The short answer is no. The diaeresis is considered obsolete by most authorities in the English language. In some cases, the diaeresis is replaced by the hyphen. Where The New Yorker writes coöperate, other publications prefer co-operate, especially outside the United States. In other cases, you’ll find it written simply as cooperate. People generally know how to pronounce these words.