Both are a form of Dimensionality Reduction but they go about it differently.
With feature selection (aka variable selection, attribute selection or variable subset selection), you select (duh) the columns in your data set that you wish to keep when training your model. An example of feature selection is Best First.
With feature extraction you end up with new (aka the Principle Components) independent variables. An example of feature extraction is A kind of Dimensionality Reduction..