The "Unique List ID" is used to identify your AWeber mailing list. This is an automatically generated ID that cannot be changed, nor would it ever need to be changed. Each mailing list that you have in your account will have its own "Unique List ID" that is exclusive to that mailing list.
To find your "Unique List ID", hover over the "List Options" tab and click "List Settings."
Then, under "Basic Information" you can find the "Unique List ID" under the "List Name."
What is the "Unique List ID" used for?
The "Unique List ID" helps AWeber to determine which mailing list subscribers should be added to through sign up forms and third party integrations. The "Unique List ID" is generally what most third party integrations are looking for when they ask for your "AWeber List Name" or your "AWeber List ID."
If you include @aweber.com after the "Unique List ID" it forms your autoresponder address. So for the above example, the autoresponder address for that list would be email@example.com.
What about my "List Name?"
Unlike the "Unique List ID," your AWeber "List Name" is purely superficial. The "List Name" is mainly for you to identify which list you are working under while you are working in your AWeber control panel.