The Components of Principal Component Analysis: A Python Tutorial Math Misery?

I recently ran a data science training course on the topic of principal component analysis and dimension reduction. This course was less about the intimate mathematical details, but rather on understanding the various outputs that are available when running PCA. In other words, my goal was to make sure that followers of this tutorial can see what terms like "explained_variance_" and "explained_variance_ratio_" and "components_" mean when they probe the PCA object. It shouldn't be a mystery and it should be something that anyone can recreate "by hand". My training sessions tend to be fluid and no one session is the same as any other.