Feature detection, also called interest point detection or keypoint detection is finding points in the image which are somehow "special". Typically that means that these points correspond to some elements of the scene that can be reliably located in different views of that scene. Examples are corners and "blobs" (centers of roughly circular regions).
Feature extraction means computing a descriptor from the pixels around each interest point. The simplest descriptor is just the raw pixel values in a small patch around the interest point. More sophisticated descriptors include SURF, HOG, and FREAK.
Matching descriptors is a way of finding point correspondences between two images.
