There is no evidence that Joseph Smith ever taught this doctrine, but it is based on things which he did teach and Brigham Young claims that it was a secret doctrine which he only taught to a few people. Scholars today, though, have concluded that the doctrine was created by Brigham Young to elaborate on and better structure some of Joseph Smith more vague ideas.
In a nutshell, the Adam-God Doctrine of Mormonism is basically that Adam is God. In the words of Brigham Young, Adam is "our Father and our God, and the only God with whom we have to do". According to Young, Adam was mortal, rose from the dead, and became a god. He found Eve on another planet, brought her to earth, and when they ate the fruit of the Garden of Eden they both became mortal.
When they had mortal children, they created both the physical bodies and the immaterial spirits of the entire human race. Then Adam and Eve returned to heaven where they rule. Later, Adam returned to earth to become the literal, biological father of Jesus.
Most Mormons seem to have accept this new doctrine, but many reject it — including some prominent, powerful Mormons like Orson Pratt. This creates enough debate within the Mormon Church that, while it is adopted as official, it's never entirely accepted like many other things. After Brigham Young dies, it's gradually marginalized and ignored. From the early 20th century on the church decides to no longer explicitly teach it, though they don't explicitly deny it either.
Instead, they argue that Brigham Young meant something radically different from how the doctrine is treated in its earliest days. This allows them to avoid saying Young is wrong, but also avoid teaching something that is so radically outside the mainstream of American Christianity. Mormon fundamentalists, though, continue to believe the Adam-God Doctrine exactly as it's first taught.