Refers specifically to the hair above the upper lip, as opposed to a full beard (あごひげ) or sideburns (ほおひげ).
He has a moustache.
I forgot to shave my moustache.
口ひげ refers specifically to a moustache, while 髭 is the general term covering any kind of facial hair.
Common mixed kanji-kana spelling; the ひげ part is often written in kana.
Full kanji spelling; 髭 is a general kanji for facial hair.
Rare kanji form; 鬚 is an uncommon variant for facial hair.
Rare kanji form; 髯 specifically refers to a beard or whiskers.
Compound of 口 (kuchi, 'mouth') and ひげ (hige, 'facial hair'). The reading くちひげ is a straightforward combination.