What is Principal component regression, partial least squares?

Principal component regression (PCR) regresses the target on the first $k$ principal components of the features (Book 4, chapter 22). Partial least squares (PLS) regresses it on $k$ components chosen for their covariance with the target: the first weight vector is proportional to $X^\top y$, and each later one is found the same way after deflating $X$ by the components already taken.