These fortune-telling cards date back to the 1400's, and Eliot seems convinced that they contain some valuable images for making sense of all that's wrong with the modern world. They're also connected to the theme of prophecy that Eliot brings up several times in the poem, also through the figure of Tiresias, the blind prophet. The tarot pack is associated in this poem mostly with Madame Sosostris, who might actually be a fraud. Nonetheless, Eliot feels that the images contained in her cards, like the falling tower or the drowned sailor, are helpful for illustrating the decline of Western society.
Lines 46-54: The cards make their first appearance early in the poem when the speaker appears to sit down with a "famous clairvoyante" named Madame Sosostris. The woman draws six tarot cards in total, which are: the drowned sailor, the Belladona, the man with three staves, the Wheel, the one-eyed merchant, and finally a card that shows a man carrying some unknown object behind his back (the meanings of the images are unpacked in the "Summary" section of this module, so head on over there for the scoop). Frankly, the speaker of the poem doesn't seem all that impressed with Madame Sosostris, suggesting that she should know how to avoid getting a cold if she knows the future so well. But on the other hand, the imagery of the tarot pack goes back to medieval England, so there's little doubt that Eliot finds something very meaningful in its cards. In fact, he thinks highly enough to use many of these images throughout his own masterpiece.
Line 55: At first, it might seem good that Madame Sosostris does not pull the "Hanged Man" card, but it turns out that the hanged man is actually a person who needs to be sacrificed before fertility and life can come back to the land; so the absence of this card is actually bad news for anyone waiting for culture to revive itself.
Lines 209-210: It's easy to miss, but the arrival of a "Symrna merchant" in this poem confirms the appearance of a "one-eyed" or immoral merchant in Madame Sosostris' prophecies. This character comes into the poem to symbolize greed and corruption.
Lines 312-321: The entire fourth section of the poem, "Death by Water," talks about the drowned Phoenician sailor, who was earlier pulled from the Tarot pack by Madame Sosostris. This figure of the sailor suggests that even when water is present in the poem, it only has the power to kill. Also, the seawater that drowns the sailor is not the same as the freshwater that promises to bring life back to the waste land. You could interpret the drowning of the sailor either as an symbol of total doom, or as a hint of hope for rebirth in the future, depending on whether or not you're a glass half full kind of person.
Lines 427-430: In the closing lines of the poem, you have both the image of London bridge falling down and that of "The Prince of Aquitaine in the ruined tower," both of which call to mind the tower struck by lightning, which is displayed on one of the cards in a tarot pack. The image represents the fall of a great figure of some kind (either individual person or civilization), and it does not offer very good news for people who want to find hope in the ending of "The Waste Land."