Replacing an on and off switch with a mechanical timer
Sometimes receptacles and switches are used to join two wires together out of convenience.  What you have is a regular switch with two terminals, but the electrician has used a push terminal and a screw terminal to join two hot (black) wires.  The push terminal and screw terminal next to it are the same electrical connection.

Lets say the top of the switch has a black wire on the screw and one in the push terminal and the bottom of the switch just has one wire in the push terminal...  To hook up the new switch, you're going to pigtail the top two wires to one terminal on the timer.  That means connecting the two black wires and a new short length of black wire with a wire nut and then connecting that short wire to the switch.  The wire currently going to the bottom of your switch will just get connected to the other terminal on the timer.