The method I use:
1. Convert the background into an editable layer either by duplicating the layer (if you want to keep the original
handy) or by double clicking on the layer title (which converts it into a new layer).
2. If possible use the magic wand to to select flat areas of background color, such as if the picture is on a white
background. I find that settings of tolerance 30 - 35, anti-aliased, contigeous works well. Holding down the shift key
on the keyboard lets you add to the selection, while ALT lets you subtract. (Many other tools have ALT, CTRL & shift
functions as well.)
3. Open the layer menu...add layer mask...hide selection. (Make sure you don't add a vector mask by mistake like I
do sometimes.)
4. Go over the layer masking with the airbrush on a hard setting, adjusting the brush size where necessary, and you can
get some incredibly accurate cutouts. black will "erase" the picture, white "draws it back it" and
various levels of grey will give you varying amounts of transparancy.
Since the layer is still there, merely made transparent with the layer mask, if you make a mistake like cutting too much
off you can easily restore it by drawing that part back in. Also you can reduce the hardness of the airbrush to create
very soft, gradual alpha masks.