What is this type of lead called and how do I wire to it correctly?
Making connections

You can use heatshrink tubing to insulate the connection and prevent the wires shorting together.



Slide a section on, out of the way, before attaching the wire, solder the wire, slide the tube over the exposed wire and heat it to shrink it.

Which connections

You don't say what this "button" is for. I don't know of a function for a push-button operating a switch with five terminals.

You can usually use the continuity function of a multimeter to work out which terminals are connected to which before, whilst and after pressing the button. From this you can work out  how to wire up the connections to whatever it controls.