You can apply shape path operations in Photoshop, combine designs in a few ways, use to create unique artwork. You can find them in the layers menu and you can also find them along the top (middle) of the control bar listed as path operations (previously set) and also the same in the properties panel.
The layer menu combine shapes feature wins out for being obvious in its use. You can select two or more objects or artworks and then use the combine / subtract etc and they work in a fairly logical way such as if you subtract then that does what it says on the tin but it does work over multiple layers.
The other approach is more a path centric one where there is a single object and it is live and it is made up of individual shapes that can still be moved by the path selection tool. Go to the top middle of the control bar and set to new layer. Before you create another, go to the drop down and select combine. Add the new one and it will be automatically combined with the existing one. Select that item and then go to the menu and select the subtract and the current object will cut away from the previous shapes. Likewise if you go to the properties panel. While they are still live, you use the selection tool as well as direct selection tool and move it around so if it is subtract then you will see more or less of the original design being removed. It is live at all times until you merge them
You can go to the bottom of the menu and select merge and the item is then merged and the subtract / combine etc is then lost and you have your finalized design.
Combine does what it says, it adds to the existing. Subtract cuts away from the existing that is why it is odd to use if you just create a design and are not cutting away from anything, it just cuts away from the entire document. Intersect just create a design from the overlap. Exclude creates the design based on the overlap but inverts it so the parts of the design not overlapping remain.