For most uses, prefer the standard spelling dialogue in both American and British English to the variant dialog (see -agogue or -logue). Prefer the variant only in computer contexts, such as in the phrase dialog box.
Dialogue, a noun only, refers to planned discourse between characters in literature, drama, film, etc. Do not use the word when referring to real-world conversations, discussions, chats, or talks—an unfortunate habit among those who try to elevate the intellectual value of their human interactions. Always prefer I spoke with my adviser and Our conversation lasted hours to I was in dialogue with my adviser and Our dialogue lasted hours.
Unless you have kidnapped and plan to torture a usage writer like myself, avoid at all costs using dialogue as a verb. We dialogued over coffee should make even the average writer squirm in discomfort.