Symbolism, Imagery, Allegory
The palantíri are the Seven Stones that the heirs to the house of Elendil used to communicate across their kingdom, sort of like an archaic version of Skype. With the fall of the kings of Gondor and the slow diminishing of Gondor's power, the Seven Stones have slowly been lost. They've either been swallowed up in natural disasters or taken over by Sauron, the greedy jerk.
Their loss by the house of Elendil is yet another sign of how far the kingdom of Gondor has really fallen. After all, along with the White Tree, the Seven Stones are one of the symbols of the power of Aragorn's family line: "Seven stars and seven stones and one white tree" (5.21.42). Having lost the stones, they have also lost their authority. And that's a big ol' problem.
The degradation of the palantíri really comes out in The Two Towers, when we learn that Saruman's use of one of the lost Seeing Stones in the tower of Orthanc has given Sauron direct access to this formerly wise leader of the White Council. Sauron corrupts Saruman using Saruman's palantír, a complete perversion of the original purpose of the stones.
Similarly, the palantír of Minas Tirith, which Denethor has been using too often, gives Sauron a tool to drive Denethor plumb crazy. Denethor's dramatic suicide with the palantír in his hands reminds us that Sauron has been able to distort things that were once used for good (e.g. the Seeing Stones, the city of Minas Morgul, etc.) to become evil. The fact that, like the palantír, both of these men started out good but wind up Very Bad, provides a parallel between the characters themselves and the stones they use to talk to Sauron. Both the men and the stones were once good and become something else entirely—something not so good.
When Aragorn takes the Orthanc stone from Gandalf, as the stone's rightful owner, we get our first sign that Aragorn is ready to introduce himself to the world as the long-lost heir of Isildur. His recapture of the palantír demonstrates Aragorn's will to redeem the tools Sauron has corrupted. They're back in good hands. Aragorn looks into the palantír and confronts Sauron directly, later telling Gimli, Legolas, Merry, and Pippin:
What do you fear that I should say to [Sauron]? Did I not openly proclaim my title before the doors of Edoras? Nay, Gimli [...] Nay, my friends, I am the lawful master of the Stone, and I had both the right and the strength to use it, or so I judged. The right cannot be doubted. The strength was enough—barely. (5.2.74)
Aragorn has built up a strong enough will that he can turn the Seeing Stone, long under the control of Sauron, into a tool to frighten Sauron himself. Catching a glimpse of Aragorn shakes Sauron's (pretty massive) confidence, and Aragorn is sure that, "To know that I lived and walked the earth was a blow to his heart" (5.2.75). It's like a good old fashioned western stare down, minus the blowing tumbleweeds.
Perhaps without this confrontation via palantír, Sauron would not have grown so focused on Aragorn and his movements, and would have turned his attention more towards Mordor and the Ring quest. The distraction of Aragorn's appearance in the Seeing Stone—and Pippin's own accidental glimpse into the Eye—gives Frodo and Sam some additional protection from Sauron's attention.