Button hole punches are metal tools used to simultaneously cut holes and a connected slit, into leather material. When working with button studs, they join leather by pushing through a hole that is slightly smaller than the stud. Hence, once through, the stud holds the button in place.
In order to have enough room to push through, there needs to be extra flexibility around the hole to accommodate the stud. This is done by cutting a slit directly connected to the button hole. The slit parts the leather, making it flexible enough around the hole to accommodate the larger stud.
Button holes can be made using a standard round drive punch, then cutting a slit below it with a knife. Alternatively, they be made with a button hole punch. This makes it much easier, with more consistent results, as all of the cutting is done with one tap of a mallet, hammer, or maul onto the button hole punch.
They are available is different sizes so the proper size can be selected based on the needed sizes of the hole and the button stud hardware that will be used.