1. Everything that exists has an identity. This identity does not depend on how people think about it or talk about it. This is also true for things like feelings or ideas. It says that a thing is the thing which it is, or "A is A." What people learn about the things that exist comes both from the identity of the things themselves and from the way that people observe (see) and think about what they have observed.
2. Reason is how a person knows that what he thinks or believes is true. A person cannot make something true just by wanting it to be true or by mysticism. Only rational, logical thinking can produce the best outcome. This means recognizing that a thing is the thing it is, and to not confuse it with things which it is not.
3. It is good to be happy, and it is good for a person to try to be happy. People should always try to improve their lives and be happy in the long term, so that they are happy now and in the future. People should not hurt others to try to be happy, but they also should not hurt themselves to try to make other people happy. People should also not make themselves less happy to help something like God. Ayn Rand called this "rational self-interest".
4. If governments or criminals take things away from people, or try to make people do things they do not want to do, it does damage to everybody. Ayn Rand thought that governments should only be able to protect people from violence, theft, fraud, and other actions that go against people's rights. This includes laissez-faire capitalism and is sometimes called libertarianism.
She first told people these ideas in her novels The Fountainhead and Atlas Shrugged. She talked about them more in The Objectivist Newsletter, The Objectivist, The Ayn Rand Letter, Introduction to Objectivist Epistemology and The Virtue of Selfishness.