Greene covers, briefly, all kinds of things: the diversity of languages in the world, what it means to have a language, the identity politics of speaking a particular language, attempts at regulating language and how they relate to nationalism. He spends a lot of time on the history of prescriptive rules for English -- think style books like Eats, Shoots and Leaves and The Elements of Style and their historical predecessors. There's also discussion on the associated hand-wringing, class issues and emotional damage inflicted by telling people that their native dialect isn't the real way to speak a given language.
So You Are What You Speak would be a good introduction to the question of "what is a linguist? what is linguistics?" for your friend who internalized the watchful eye of your high school English teacher and yells at people about their grammar and diction on the Internet. If anything, I think Greene gives too much credit to language prescriptivists by suggesting that there is some kind of meaningful debate going on between sticklers and, y'know, scientists trying to describe language in the world.
I would have liked to see more examples from outside the Western-European world. Greene spends most of the book talking about English and French, with some bits about the Brazilian Portuguese language academy (which I didn't know was a thing). Come to think of it, more concrete examples about the socio-politics of different English dialects would have been good too. But it's not that long of a book.
So if you've been hanging out in a Linguistics department -- or just reading Language Log -- and laugh when people despair loudly that kids these days are destroying the English language, you may not need to read this book. But you might want to give it to your relatives.