| I don't at all understand that business about Unicode (later - oh, that writeup's been removed), so I'd better write down what I do understand. Ligatures and digraphs are letters. A diphthong (sic, not dipthong) is a sound.
A digraph consists of two separate letters, like SH or TH or CH or EA or OI or OU or AE, that represent a single sound.
A ligature consists of two letters joined together, either in handwriting or in print. The two ligatures I can write here are æ and œ. (And I can't even guarantee that your browser will show œ or oe as joined up.) These are relatively rare, of course.
There are five ligatures in a normal typeface. Any properly printed book will show the combinations ff, fi, fl, ffi, and ffl as fused. They are a single piece of type. There is no dot on the i in the combinations fi and ffi. (I have an idea some superior word-processors can now manage to do this. - ariels mentions that LaTeX can.)
German has a word Sauerstoffflaschen 'oxygen flask' which uniquely contains a group fffl, representing a problem for type-designers. Thanks to FordPrefect for this bit of trivia, and see that node for a solution.
Some fancier typefaces have ligatures for ct, st, and ss, with an elegant loop joining the two.
The German eszet symbol ß (ß) apparently derives from a ligature sz. (The modern eszet represents ss, and looks like it comes from ss - but in the old Fraktur or Black Letter type it's clearer that it's s + z, i.e. es + zet.) Possibly it was originally a ligature of long s and round s and was reinterpreted in Fraktur.
A Black Letter capital F looks like lower-case ff: see that node for how this has caused some surnames to be written as e.g. ffrench or ffoulkes.
A diphthong is a combination of two vowel sounds, as distinct from a pure unchanging vowel. In a phonetic script it is natural to represent a diphthong with a digraph; for example English OI in 'boil'. This consists of an O sound followed by an I sound. But because of the history of English over a thousand years, you often get single vowel letters represent diphthongs (as in 'go'), and digraphs representing simple vowels (as in 'wood'). | 
