A trema (from the Greek τρῆμα trêma; plural tremas or tremata) is a diacritic consisting of a pair of dots ( ¨ ) placed over a letter, most commonly a vowel. (When that letter is an i, the diacritic replaces the tittle: ï.) The trema is usually used to denote one of two distinct phonological phenomena: diaeresis ( ), in which the trema is used to show that a vowel letter is not part of a digraph or diphthong; and umlaut ( ), in which the trema illustrates a sound shift.
The diacritical mark is itself commonly referred to as either a diaeresis or umlaut, depending on what role it is fulfilling. The two uses evolved separately from one another, with the association of the trema with the diaeresis being considerably older. In modern computer systems using Unicode, umlaut and diaeresis are identical: ‹ä› represents both a-umlaut and a-diaeresis.
The diaeresis indicates that two adjoining letters that would normally form a digraph and be pronounced as one are instead to be read as separate, either as a diphthong or as two distinct vowels in two syllables. To put it simply: a diaeresis indicates that a vowel should be pronounced apart from the letter which precedes it. For example, in the spelling coöperate, the diaeresis reminds the reader that the word has four syllables co-op-er-ate, not three, *coop-er-ate. This is uncommon in English, as it is optional; The New Yorkers use of the mark is idiosyncratic. Languages such as Dutch, French and Spanish, however, make regular use of it. By extension, the word diaeresis also designates the diacritic when used to denote similar distinctions, such as marking the schwa ë in Albanian.
"Um"+"laut" is German for "around/changed"+"sound". It refers to a historical sound shift in that language. In German, the umlaut diacritic is found as ä, ö and ü. The name is used in some other languages that share these symbols with German or where the Latin spelling was introduced in the 19th century, replacing marks that had been used previously. The phonological phenomenon of umlaut occurred historically in English as well (man ~ men; full ~ fill; goose ~ geese) in a way cognately parallel with German, but English orthography does not write the sound shift using the umlaut diacritic. Instead, a different letter is used.
Historically, the diaeresis mark or trema is far older than the umlaut mark.
Historically, the umlaut mark is far younger than the diaeresis mark, and has unrelated origins, though it has been speculated that an awareness of diaeresis might have influenced the final written form of the umlaut.
Originally, phonological umlaut was denoted in written German by adding an e to the affected vowel, either after the vowel or, in small form, above it. (In medieval German manuscripts, other digraphs could also be written using superscripts: in bluome (“flower”), for example, the <o> was frequently placed above the <u>, although this letter survives now only in Czech. Compare also the development of the tilde as a superscript ‘n’.) In blackletter handwriting as used in German manuscripts of the later Middle Ages, and also in many printed texts of the early modern period, the superscript <e> still had a form which would be recognisable to us as an <e>. However, in the forms of handwriting which emerged in the early modern period (of which Sütterlin is the latest and best known example), the letter <e> had two strong vertical lines, and the superscript <e> looked like two tiny strokes. Gradually these strokes were reduced to dots, and as early as the 16th century we find this handwritten convention being transferred sporadically to printed texts too.