The White Tree is the symbol for Gondor. When Arwen makes Aragorn his war banner, she embroiders "a white tree flowered upon a sable field beneath a shining crown and seven glittering stars" (6.4.44). So the white tree represents Aragorn's hereditary right to the throne of Gondor. Plus, it just sounds so pretty.
With the slow disintegration and eventual disappearance of Aragorn's family line from the throne of Gondor, the "Tree in the Court of the Fountain" (6.5.109)—as in, the actual, physical White Tree, on which the symbol of Isildur's house is based—withers and dies. So the White Tree of Minas Tirith is clearly connected to the health of Gondor's kings, and to the wellbeing of the kingdom in general. When Gondor loses its king and comes under the power of the line of Stewards, the White Tree dies out.
Luckily, when Aragorn comes back to Minas Tirith, Gandalf leads him to the mountain behind the city. There, Aragorn finds a little baby White Tree: "Lo! here is the scion of the Eldest of Trees!" (6.5.112). And Aragorn replants this tree in the Court of the Fountain. Its health and rapid blooming proves that everything is hunky dory in Gondor again, now that Aragorn is back in business. In fact, when the Black Gate first falls and the Eagle comes flying to Gondor to spread the news, the proof the Eagle offers that things are going to get better is that the White Tree will come back to life:
And the Tree that was withered shall be renewed,
and he shall plant it in the high places,
and the City shall be blessed. (6.5.49)
But the White Tree doesn't just symbolize Aragorn's family or the health of Gondor in general. It also represents the link between the Men of Númenor (from whom the people of Gondor are descended) and the origins of Middle-earth in the lands of the Utter West and the Blessed Realm of the Valar (the Lords and Ladies who have made Middle-earth). That's a fancy, Tolkienesque way of saying that this tree is a symbol of Gondor's roots, which are long and, well, pretty powerful.
This baby tree that Aragorn finds on Mount Mindolluin (the mountain above Minas Tirith) is a descendant of that first tree that Isildur brought with him from Eressëa, the lost island of the men of Númenor, which in turn comes from the Two Trees Laurelin and Telperion, which gave light to the Realm of the Valar before they were destroyed by Morgoth in the Silmarillion. We know this is a pretty important tree because Gandalf bothers to trace its descent line: "Verily this is a sapling of the line of Nimloth the fair; and that was a seedling of Galathilion, and that a fruit of Telperion of many names, Eldest Trees" (6.5.113). That tree has a longer line of ancestors than we do; we can tell it's insanely important, symbolically speaking.
The return of the White Tree to Minas Tirith proves that Aragorn has restored Gondor's former glory as the home of the Men of Númenor. Sauron, a servant of the original Great Enemy who destroyed the Two Trees in Valinor, has been driven permanently from Middle-earth. The replanting of the baby tree descended from those original Two Trees shows that we have come full circle: the creature who destroyed the Two Trees in the first place has been conquered and Good (represented by that new young tree) has won out in Middle-earth.