That's a very big question. Let me tackle the fundamental issue.
Diacritics are marks that essentially change the character to which they are applied. Common examples are
ê. Greek is full of them.
Many of these marks are the result of evolutionary alphabet development.
Ü, for instance, is really a digraph (two characters that make a new character when used together like the Spanish
ll) that has evolved from shorthand into a diacritic. The
¨ (umlaut) was originally the letter
e used to change the sound of other vowels. In blackletter/fraktur faces (used in Germanic manuscripts of the day) the e was reduced down to it's strongest elements (the vertical strokes) and placed above the vowel to be modified. In modern times it evolved from two broad nib pen strokes (parallelograms) to two circles.
If you're trying to access them from your keyboard, it depends on your keyboard localisation. In OS X you can take a look at the diacritic key sequences by bringing up the keyboard viewer.