Sometimes, when digraphs do not represent a new phoneme, they are a relic from an earlier period in the language's history when they did.
Transliteration makes extensive use of digraphs.
There are three kinds of digraphs: sequences, reversals (really a special kind of sequence) and doubled letters.
This is a group of two letters, both of which are different. English is full of these, for instance the ch and th used in English spelling. Ch is an affricate and rarely a fricative, th is one of two fricatives, and sh is a fricative. Examples from other languages include French's ch, which has the value of the English sh, ou, which is the W sound, gn, which is the palatalized n /nj/, and Spanish's ch, which has the same value as in English.
Reversals are sequences in which both possible orders of letters are common enough to be digraphs. Er and re in English are the most common example.
These have both letters the same. In some languages these indicate a stressed syllable or a new sound, and in some cases they are just part of the spelling convention. Ll is the most common in English, though it represents no new sound, but that is not the case in other languages; Welsh's ll is a voiceless lateral, and in Spanish it is a palatalized l /lj/ (Castilian only) or else a palatal fricative. Ee and oo are common examples from English. Rr in Spanish indicates a trill, and forms minimal pairs with the single r. Italian's zz represents the affricate /ts/.
See also:
Sequences
Reversals
Doubled Letters