Q: So how DO the members of the Order of the Phoenix communicate with each other?
A: I was surprised that this particular question won the poll, because the answer (as I've already said) can be found in an already-published book (Goblet of Fire), whereas the other two questions related to book six. But perhaps I was influenced by the fact that I knew the other two questions had interesting answers – and, of course, you will shortly know the answers to those questions anyway!
Members of the Order use their Patronuses to communicate with each other. They are the only wizards who know how to use their spirit guardians in this way and they have been taught to do so by Dumbledore (he invented this method of communication). The Patronus is an immensely efficient messenger for several reasons: it is an anti-Dark Arts device, which makes it highly resilient to interference from Dark wizards; it is not hindered by physical barriers; each Patronus is unique and distinctive, so that there is never any doubt which Order member has sent it; nobody else can conjure another person's Patronus, so there is no danger of false messages being passed between Order members; nothing conspicuous needs to be carried by the Order member to create a Patronus. - jkrowling.com
And, as many of you have deduced, Dumbledore's Patronus is indeed a phoenix.