I've pulled out the single pole switch and there are three black wires, two of which are connected to the same screw and one connected to the screw on the opposite side of the switch. Both screws appear to be brass and there is nothing connected to the odd colored screw on the switch. When I put in the new dimmer switch, do the three black wires get connected in the same way on the dimmer switch leaving the odd colored screw empty?
Yes your wires get connected the same way (but not 2 wires under 1 screw, unless it is a clamp designed for 2). If you are connecting the 2 wires and they are wrapped around the screw only 1 wire can be bone that way so a pig tail is needed a short pice of the same gauge wire wire nutted with the 2 wires then to a screw terminal, or better yet if the new switch has leads just tie the lead to the 2 with a wire nut and you are good to go.
Two separate wires should never be attached to the same screw. Are you sure they are two separate wires and not a single wire with the insulation stripped off in the middle?
If it is two separate wires, a short black pigtail wire should be used. One end of the pigtail attaches to the screw on the switch and the other end is joined to the two black wires in the box using a wire nut or other suitable connecting device.
Is the "odd colored" screw green? This is the ground connection. Inside your junction box, there should be two or more bare copper wires joined together. Since it sounds like you old switch wasn't grounded, you will need to create another pigtail (using bare copper wire) and join it to the other bare copper wires in the box and the other end connects to the ground screw on your new switch.
A picture showing what is in the box would be very helpful to make sure everything I said above is accurate. A lot depends on where in the world you are located and the age of the wiring.