The “o with two dots” is known as the diaeresis or trema, and it is a diacritical mark that consists of two dots placed over a vowel. In English, it is most commonly used in loanwords from other languages, particularly German, to indicate that two adjacent vowels should be pronounced separately, rather than as a diphthong.

The diaeresis is also used in other languages, including French, where it is used to indicate that an e should be pronounced separately from the preceding vowel, and in Hungarian, where it is used to distinguish between two different vowels that would otherwise be pronounced the same.
The diaeresis has a long history, dating back to Ancient Greek, where it was used to indicate that two adjacent vowels should be pronounced separately. It was later adopted by Latin and other European languages, where it was used to indicate that two vowels should be pronounced separately or to distinguish between two different sounds that might otherwise be confused.
In English, the diaeresis is most commonly used in loanwords from German, such as “über,” “höflich,” and “Sänger.” Without the diaeresis, these words might be pronounced as “oober,” “hoflich,” and “Sanger,” which would be incorrect. The diaeresis indicates that the two adjacent vowels should be pronounced separately, as in “oo-ber,” “hof-lich,” and “San-ger.”
It is worth noting that the diaeresis is not the same as the umlaut, another diacritical mark used in German and other languages. The umlaut consists of two dots placed over a vowel to indicate a change in pronunciation, usually from a back vowel to a front vowel. For example, in German, the word “Mädchen” (girl) has an umlaut over the “a” to indicate that it should be pronounced as a front vowel, rather than a back vowel.
In French, the diaeresis is used over an “e” to indicate that it should be pronounced separately from the preceding vowel. For example, in the word “naïve,” the diaeresis indicates that the “i” should be pronounced separately from the preceding “a,” rather than as a diphthong. Without the diaeresis, the word would be pronounced as “nayv,” which would be incorrect.
In Hungarian, the diaeresis is used to distinguish between two different vowels that would otherwise be pronounced the same. For example, the word “öreg” means “old,” while the word “oreg” means “watchdog.” The diaeresis over the “ö” indicates that it should be pronounced as a different sound from the “o” in “oreg.”
In conclusion, the diaeresis is a diacritical mark consisting of two dots placed over a vowel to indicate that it should be pronounced separately from the preceding vowel. It is most commonly used in loanwords from other languages, particularly German, to indicate that two adjacent vowels should be pronounced separately, rather than as a diphthong. The diaeresis has a long history and is used in a number of different languages, including French and Hungarian, where it is used to indicate different sounds or to distinguish between similar sounds.