The diaeresis is an artifact of the English language that, for most purposes, has fallen out of use, unless you are The New Yorker. The diaeresis fits atop a vowel to indicate that it must be pronounced separately from the previous sound. For examples, Zoe would, under regular English customs, be pronounced to rhyme with row, but the diaeresis informs us to pronounce the e separately. The same goes for cooperation, reelection, zoology, etc. In the past, the second vowel received a diaeresis. Unfortunately, history and laziness took over and we got spellings like co-operation, re-election, and eventually we gave up on all that, and now spell them without any artifacts. The same goes for spellings of Chloe and Zoe. Except for a few magazines, books, and the random nerdy blog post, you won't hear or have to worry about those pesky English markings.
But someday, when you're middle-aged and at some drab party with professors and their wives, drop this tidbit and you'll feel just dandy.