We know from Book 1 that Harry and Voldemort's wands share the same core – phoenix feather. (In Goblet of Fire, we find out that the phoenix in question is Dumbledore's dear Fawkes.) The shared wand core has huge plot significance in Goblet of Fire, since that's what triggers the Priori Incantatem effect that allows Harry to get away from Voldemort. It's also symbolically significant in the same way Harry's curse scar is: Voldemort and Harry are linked by their magic. That's how we know that Voldemort is going to keep coming up as part of Harry's destiny, no matter how many times Harry escapes or delays their inevitable showdown. What's more, phoenixes are known for coming back to life even when everything seems lost. This is something that both Harry and Voldemort (regrettably) have proved able to do.
In general, wand cores say something about the personality of the wizard or witch who carries it. Fleur Delacour's wand uses hair from her veela grandmother's head. Mr. Ollivander comments that veela hair "makes for rather temperamental wands" (18.172). And Fleur is pretty darn temperamental herself, with her 180-degree turnaround on what she thinks of Harry. Cedric's wand comes from the tail hair of "a particularly fine male unicorn" (18.176), and Cedric is a particularly fine young man. All the associations with unicorns are good: they're brave and pure of spirit. Cedric seems to share those qualities. Last but not least, Viktor Krum's wand contains dragon heartstring and is "quite rigid" (18.183). He is also stiff, awkward, and somewhat dangerous, but he's brave and strong, a lot like a dragon.