What are the two dots punctuation marks in the word noel called they are usually put over the e?
It's called an "umlaut" and it's not punctuation.
One of the big differences between English and other European languages that use the Roman alphabet is, we avoid using diacritical marks and other languages use them a lot. Diacritical marks affect the way a word or letter is pronounced, whereas punctuation marks affect the flow or termination of a sentence. (The apostrophe is both punctuation and diacritical, and is the only diacritical commonly used in English.)
The "two dots" you ask about are known by a couple of different names: diaeresis (for Greek words), tréma (for French), and umlaut (German). Each of these makes a slightly different sound. The diaeresis over a letter (always a vowel) means that the letter is pronounced separately from the letter right before it. Coöperate, for example. The umlaut, used in German, changes the sound of the vowel it is over, sometimes the U sound to a "yu" sound, or an O to an "eh."
When Americans use it, it's usually to make a name seem more exotic (Chloë Sevigny) or creepy (Mötley Crüe), but it can come off as a bit pretentious.