What you're referring to is called image cutting and most of the available image editors have this option built-in and ready to use. There are professional image editors like Photoshop, but you can achieve this with the help of Photoscape, for example.
Download: http://www.photoscape.org/ps/main/download.php
Install the software, load the image and then you can use the Cut feature to remove a part of the image to paste it somewhere else.
