Just select on object, area you want with Quick Selection Tool. If you needyou can easily adjust/refine selection using Select&Mask Command (refine Edge in previous version PS).
When you are happy with your selection prec Keyboard Shortcut - CTRL+J / CMD+ J and go to Layers Panel (to see Layers you can precc F7 key on your Keyboard). Finally Clik to close Eye icon seen on left your Background Layer to turn it off.
Then seave your image as png file
Instead of CTRL + J command you can use Move Tool (V) and just drag selection into antoher document.
Be carefull with dragging it's a little be tricky
The selected area will be copy/pasted into your second picture with the correct transparency. You need however to to a good and precise work with the selection, otherwise the result will not be good. So pasting is not the problem, but creating a correct selection is, depending on the picture.