To authorize your ADE, menu/Help/Authorize Computer.
To authorize your eReader, first authorize the computer then with the device plugged in, menu/Help/Authorize Device.
Sometimes you get confused authorization: more below but in summary in that case deauthorize and reauthorize
deauthorize computer: ctrl-shift-D (cmd-shift-D on Mac)
deauthorize device: ctrl-shift-E (cmd-shift-E on Mac) while plugged in and running ADE
Sometimes ADE gets its registration/activation confused and in a semi-authorized state.
Uninstalling and reinstalling does not help.
Unfortunately, it often then gives misleading error messages about what is wrong.
A common incorrect message informs you that the ID is already in use on another computer and cannot be reused.
This can often be resolved by completely removing any authorization using ctrl-shift-D to the Library screen on ADE (cmd-shift-D if on Mac).
Restart ADE, and then reauthorize with your (old) Adobe ID.
In extreme cases on the mac, the following extra step has helped some people. Navigate to /Users//Library/Application Support/Adobe/Digital Editions and drag the activation.dat file to the trash. If you are using 10.7, see Access hidden user library files | Mac OS 10.7 Lion. http://forums.adobe.com/thread/1265248?tstart=0