Switching new dimmer for old dimmer
Assuming the dimmer is similar to the Lutron dimmer in this question:
The new dimmer can be used as a regular switch or a 3-way switch.
For a regular switch, which it sounds like the way your old dimmer was connected:

Black - Hot
Red - Switched Hot
Green - Ground
Red/White - Not used - cap with a wire nut

For a 3-way switch:

Black - Hot
Red - Switched Hot/Traveler 1
Red/White - Switched Hot/Traveler 2
Green - Ground

The second switch for a 3-way setup should be a regular switch, not another dimmer.
Wiring a 3-way switch can be a bit confusing. Search for other information on that (plenty on DIY StackExchange, or Google it) and then just treat the dimmer as one of the 3-way switches.