To demonstrate these first select the polygonal lasso. Set anti-aliasing off and feathering to zero. Make a narrow selection with some diagonal lines.
Switch anti-aliasing on and add another selection.
Set feathering to, say, 3 pixels and add another selection.
Now use the paint bucket to fill the selection with a bright colour.
Zoom in to see the effect.
With a normal selection all pixels are either selected or not. In close up you can see that diagonal lines have jagged edges.
Anti-aliasing smoothes these jagged edges by partially selecting pixels along the edge.
Feathering partially selects pixels on either side of the edge to produce a faded effect.
When working with photographs you should always have anti-aliasing switched on.
If you are selecting a particular feature, e.g. a face, the selection will usually look a lot more natural if you use a small amount, say 3 pixels, of feathering. This is particularly true if you are cutting and pasting, as we shall learn later.
If you want to apply a general effect, say to darken the top of the sky, use a large amount of feathering to merge in the effect.