Photoshop has many kinds of tool to make copy (or cut) and paste, such as magic wand, pen, lasoo, etc. For simple cut & paste, use the simple selection tool, see my picture the left arrow. Click on it then make the selection on the picture you want to copy/cut. Then click copy (or cut) and put your cursor in the place you want to paste --> click paste.
Slice tool is more for making webpage, not for cut and paste.
For detail, you can click "help" menu in the Photoshop.
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