We call this a cut-out image.
I don’t know whether you know the best way to do this but I’ll tell you anyway just in case - I am presuming you’re using PhotoShop
Use the pen tool to draw a path round the image - always draw your path just inside the image you’re cutting out.
Save the path
In the path palette go to Clipping Path and select the path you’ve just saved.
Save as EPS.
You don’t need to get rid of the background this way so your image is still intact for future use.
