Some guy I don't like told his friend I was acting all demure. What does that mean?

Give that guy some points for vocabulary! He could have said you were putting on a front by faking modesty, which would be one way to describe demure behavior. You also could be considered demure if you're modest or bashful and not pretending.

Demure often appears in works of literature — both classic and modern.

In Vanity Fair, William Makepeace Thackeray writes,

She cast down her eyes demurely and smiled as they started away from her; Emmy scurrying off on the arm of George (now grown a dashing young gentleman) . . .

In another chapter, he writes,

Both of them were rather agitated, and thinking of old times. He about Eton, and his mother, whom he remembered, a frigid demure woman . . .

From Sinclair Lewis's Babbit:

. . . Babbitt sat creakily back in his revolving desk-chair and beamed on Miss McGoun. He was conscious of her as a girl, of black bobbed hair against demure cheeks.

The same author also wrote, in his novel Main Street,

. . . at two minutes to five a town of demure courtyards and welcoming dormer windows had been erected, and at two minutes past five the entire town was as flat as Babylon.