The mark with two dots can refer to various things.
Dieresis (Greek word we use in English) is a phenomenon where two vowels don’t merge together to make a new sound. Co-operate, coöperate, or cooperate. You could also shove a glottal stop in there but this is much less extreme. The diaresis marker isn’t used much in the USA. Another common example is naïve.
Umlaut (German word) is the opposite phenomenon: the first vowel sound is inflected by the following, so in German Apfel (apple, pronounced UPfell) becomes Aepfel (apples, EPPfell), instead commonly written Äpfel instead.
People usually refer to the marks by the term in their own language (umlaut, diaresis, caron, accent…). I say Umlaut in German (which doesn’t mark the phenomenon, you just have to know) and diaresis in English though I’m talking about the same symbol.
But diacritics are used in no systematic way in different languages anyway (this is true of the letters themselves too: letters like S, W, V and Z have different pronunciations in English and German, for example). Ppl just needed ways to say “the sound here is similar to the sound of this letter but not really the same” or “when you say this letter also do something else” (like, say, add stress for languages that have that).
English mostly dispenses with them, sometimes adding a letter (g vs gh) but mostly having a very casual relationship with the sound at all. You simply have to know that cooperate isn’t pronounced like chicken coop. I personally think that’s a good thing.