Divination by tea leaves is know as "tasseography." It is believed to have originated with the Chinese. For traditional tasseography one uses a teacup (having a wide mouth, sloping sides, and a plain interior) and tea leaves (not tea bags).
After the tea is drunk, the dregs are swirled, the cup inverted and the liquid allowed to drain. The patterns formed by the tiny clusters of leaves are then "read," much as one would see pictures in clouds or inkblots and assign meanings to them. For example, a picture of a bell signals "unexpected news," a daisy "happiness in love," and so forth.
Pictured is an advertising card for Chase & Sanborn Tea showing some ladies engaged in tasseography. Copyrighted 1892, it is captioned, "What vision, dear mother, in your cup do you see? The whole world drinking Chase & Sanborn's Coffee and Tea." To its right is a postcard, dated 1908, stating that "Good Fortunes" may be "told by Tea or Coffee grounds" and (as pictured) clusters of squares indicate "Peace and Happiness."