We've been slowly learning about the odd and powerful Expecto Patronum spell since Harry was first taught how to cast it in Book 3, The Prisoner of Azkaban. While at first, a wizard's Patronus seemed merely to be a kind of protective measure – the only sure-fire way to combat the dreaded despair brought on by dementors – they appear now to be more than that. As we've seen in this book, wizards can send their Patronuses to communicate with others far away (both Kingsley and Mr. Weasley do this amazing trick early in the book), and they can also accomplish tasks for the wizard; witness, for example, Snape's deployment of his doe Patronus to guide Harry to Gryffindor's sword. Now that's one handy Patronus.
Speaking of Snape, we also learn in this book that Patronuses can communicate more than we think about a person. In earlier books, the Patronus seemed to be a kind of manifestation of the wizard's personality, or something deeply connected to that person. For example, Hermione's otter Patronus or Ron's energetic terrier both seem merely to demonstrate personality traits – perhaps Hermione's cleverness and her affectionate nature, or Ron's, er, doggedness (ba-dum-ching!) and determination. Similarly, Luna Lovegood's is a hare, showing her rather hare-brained (but quick) persona.
Harry's Patronus has always been slightly different, though – it's a stag, James Potter's Animagus alter-ego, which represents the protection and love of a father Harry never knew. The stag Patronus is a kind of connection to Harry's past, indicating that the Patronus can also be shaped by relationships, and by the people or things that give the spell-caster strength. This idea is emphasized even more in Book 7. Both Lily Evans and Severus Snape's Patronuses (Patroni? What is the plural, anyway?) represent the loves of their lives in slightly different ways. Lily's is a doe, the natural companion to James's stag; Snape's is also a doe, but it functions kind of the way that Harry's stag Patronus does, serving as a reminder of the person he loved and lost, Lily. Rowling explained that a "Patronus often mutates to take the image of the love of one's life (because they so often become the 'happy thought' that generates a Patronus)" (source).
A person's Patronus is thus always an important marker of their personality, their relationships, and their motivations, so whenever you see someone's Patronus mentioned in any of the Harry Potter books, pay close attention – it will always be significant!