Remove white areas

In addition to what has been said, the unique layer called Background cannot hold transparency. Rename the layer to make it a regular layer.

JPEGS can't have transparency, so save in another format such as a PSD or TIFF or PNG, depending on where the image will be used.

~ Jane

Perfectly removing all white 

It's difficult to say without seeing your image.   This is one method but there are others.

1. Select white with your Magic Wand tool. 
2. Refine the selection in Quick Mask mode.
3. When all white parts are selected (surrounded by marching ants), hit the Backspace key to remove.
4. Use Ctrl / Cmd + D to deselect.

A screen shot may help us to see the issue.