Hi
It depends on how you want to use the image after your crop it, but in Photoshop you would create a clipping path that would select the image but not the background and save it as a Tiff or native Photoshop file.
When pasted only the image shape would show.
It also depends on file type, for example PNG file will display with a transparent background and the same file saved as a .JPG will not.
So if you save a .PNG file that doesn't have a background, i.e. it's on a layer and doesn't have anything behind it, then it will display as only the shape you had on the layer.
It would help to know how you are going to use, or display an image that doesn't have a background?
Most image software will show a white background behind an image even it there is no background saved with the file.
It has to put something there, Photoshop is an exception, it will place the image on a layer, and show a blue checkerboard behind the image to show there is no background.
Do you mean you want to paste the image over another photo and have only what you are pasting show?
There are a lot of way to do that so it all depends on what how you are going to use it.
Mike