Conversation analysis (CA) focuses on the language, practices and competencies by which people accomplish social actions to create and understand ordinary social life. CA uses naturally occurring data, examining micro-detailed transcriptions from recordings of ordinary interactions. This paper highlights some principles, methods, and insights of CA. We consider a short segment of transcribed phone conversation in which one participant proposes a social visit to the other. We see just how the talk develops as it does, and examine the details of language-in-use that the participants themselves draw upon to construct and make sense of what it is they are doing, of what is going on. How does a 'proposal' for a social visit arise from a course of talk, and how is it fitted both to its recipient and to the moment it occurs? How is the proposal understood to be 'unsuccessful'? We show how turns at talk always emerge and are understood within the rich sequential context of a developing trajectory of interaction that is jointly developed by participants. Linguists are well placed to study communication as naturally occurring talk, and such study can in turn inform linguistics by developing knowledge of the nature and use of language.