As a genre label, originally applied by the notoriously mean British music press (see NME) to refer to sickly-sweet indie pop music like Heavenly and Talulah Gosh. In the States, the best know sources are Slumberland Records and K Records.

Twee is actually an older term than the British pop music press. Originally used in reference to opera, the term describes something that is consciously cute. An example in opera would be pretty well anything by Gilbert and Sullivan (such as Pirates of Penzance).