Published in 1952, the book is based on a series of wartime radio broadcasts on Christianity. By "mere" Christianity Lewis means "common or central" Christianity. He says in his preface he made an effort to discuss those ideas that are without denomination, and he sent the second part, "What Christians Believe" to Anglican, Methodist, Presbyterian and Roman Catholic clergy asking for criticism to make sure he was right on the mark. He says he finds this particularly important in addressing his "unbelieving neighbors" and the book is an apologia, or argument for and defense of what he considers the Christian core beliefs--and as such particularly addressed as much or more to atheists as to believers. As an atheist myself who was raised as a Catholic, I can't say I find him convincing--or even enlightening about Christian belief.
The work is divided into four books. The first, "Right and Wrong as a Clue to the Meaning of the Universe" is an argument for Natural Law ethics and for it as a proof of God's existence. Anyone who has taken an introduction to philosophy course knows there are three basic arguments for the existence of God: ontological (that by definition God must exist), cosmological (some great being must have created a universe) and teleological (a watch means there must be a watchmaker, a universe shows complexity and purpose that requires a designer). Lewis' argument is a variety of the teleological argument. All human beings have a common core morality. Such universality means a God must have designed it so. Moreover, it must be God's design, because we have no selfish interest in being moral. Lewis' argument from intelligent design is subject to all the refutations you can find in any philosophy text.
As for morality being something inconvenient and not selfishly "good" for us--I'd dispute that. One commonplace of market and bargaining theory is this. If you're only going to deal with a person once, it's in your best interest to cheat them. But that's not how markets work. We all have to deal with the same people again and again, and it's in our best interest therefore to treat people fairly and honestly. And on a personal level, well think of Scrooge. Who was happier? The miserly Scrooge eating thin gruel and going "Bah, Humbug?" Or the reformed, generous and benevolent Scrooge who became a much loved family member and friend? Virtues aren't something inflicted upon us by authority. That doesn't mean they're easy to practice or don't call for discipline and short-term denial in exchange for long-term payoffs, any more than it's easy to follow a diet even though it might be best for our health. Arguably virtues are the habits of mind and action that best helps human beings to flourish. And that doesn't need a God. As Lewis admits, they're fairly universal and accessible to all.
Even if you believe in God though, there's a huge gap between that belief and belief in the specific doctrines of Christianity. The second part "What Christians Believe" tries to jump that gap and can be summed up this way. Jesus claimed he could forgive the sins of others. Anyone claiming to do so if not God must be a "lunatic or fiend." The Jesus of the Gospels is neither lunatic or fiend, and therefore must be God. This begs the question. I actually don't think there's anything in the picture of Jesus in the Bible that proves that he wasn't delusional or (less likely in my opinion) claiming a connection with God he didn't have for personal ends. History is filled with would-be messiahs from Mohammed to Joseph Smith. There's no reason to believe Jesus or Mohammed or Joseph Smith has a special connection with God except the believer wills it so. And there's a third alternative. That Jesus was neither lunatic nor fiend but was misunderstood and misrepresented by the fallible men that wrote the works in the New Testament.
And speaking of the New Testament, one problem I have with Mere Christianity is if you're going to lay out what it is Christians believe, I think you should carefully parse your sources. I can't recall Lewis ever quoting the Bible, and he definitely mixes things that are core Christianity with elaborations infused with Greek and Roman philosophy. I defy anyone to show me anywhere in the Bible that discusses the "cardinal virtues." These come from the Pagan Greek philosopher Plato's Republic and were adopted into the Christian tradition though theologians such as Augustine and Thomas Aquinas familiar with classical philosophy. I would have liked to know what came from what Jesus reportedly said, what came from other New Testament sources and what is simply received Christian tradition. I get Lewis' purpose in trying to argue from the ground up for Christianity for the ordinary person, and not wanting to load it up with a history lesson. But if the book doesn't work for me as a good argument for Christianity, it's also not in my opinion necessarily the best place to explore just what is Christianity.
The third part "Christian Behavior" is where Lewis discusses those cardinal virtues, and the theological virtues (which come from Paul in the New Testament) and such issues as "Sexual Morality." I actually found that particular chapter fairly insightful and full of common sense. But as with the other talk of morality, where it makes sense, I don't think you need to bring in Christianity to argue for it, and when it doesn't make sense (such as Lewis' argument for the man being the head of the household in a marriage) it just focuses attention on a facet of Christianity which I don't think holds it up in an attractive light. For me, the last part, "Beyond Personality" about theology is about as meaningful as an argument about how many angels can dance on the head of a pin. For a Christian, Lewis' argument for the truth of the trinity may be profound. For me it was just... well nonsense. And non-sense.
C.S. Lewis is a very lucid writer and his arguments thought-out and presented well if you accept certain assumptions. I can imagine for many readers he has a lot to offer, but I can't say this did much for my understanding of Christianity, and despite Lewis claiming he aimed it at least partly at nonbelievers, I think this might be more for those Christians who want to think about the fundamentals of their faith and what it demands from them.