Other than WHITE and GREEN there are NO standard color coding for electrical wiring. It's more important to look at the house wiring and then READ the instructions that came with your dimmer.

Equipment grounds are always green or bare. It's safe to always connect these together.

White is the grounded (i.e., neutral) conductor USUALLY. However, in older houses it may be a power feed for a switch loop.

Two switches controlling the same light are called "three way." One terminal is a common which goes to either the lamp or power. The other two are the "travellers" which go to the other switch.

What are the wires connected to the other switch (there should be three wires in addition to the green/bare).

Then you'll have to tell us what the model dimmer you bought (or at least read the instructions for yourself). Not all dimmers can be connected to a regular threeway by the way.

What you want to do is hook up your 2 reds in the place where the two blacks were off the old switch and hook the black up to the wire that the red from the old switch was hooked up to. It looks like the new switch just reversed the colors. This will work as long as the switch is a three way, which it sounds like it is.

Thanks for the help...will give it a go and report back.