Sheepdog,
If you are trying to setup remote IP access and you need help with DID's then you should definately consult a professional because you will need to have advanced knowledge of setting up a network, a VPN and administering IP stations.
A DID number is simply a direct inward dial number on your switch. For example your direct line phone number would be a DID number.
To find out what your DID range is, go to your switch and type in the command: display dialplan
Anything that says extension will be an extension on your switch and at least some of them will be DID's, find out what the first digit is in your Dial Plan that is DID and then you can search for an available one.
If you have ASA you can us the command called "Find an available extension" and it will automatically find the next one for you.
You will need to either create a new IP station or duplicate one that is already existing with the new DID number you found.
If you have an existing network and VPN then you can either purchase IP hardphones or install IP Softphones on their computers - (make sure you have your Avaya RTU license for your current softphone model)
If you need more help, I will suggest that you break up this request into several categories that you need help with and post them in the "Avaya (Lucent): Definity Systems" forum"