I haven’t checked email yet but wanted to chime in here since it’s a pretty universal answer…
SketchUp for Windows will have a Serial Number and Authorization Number.
SketchUp for OS X will have a Serial Number and Key.
For all intents and purposes, an Authorization Number is a Key and you may bump into one where you’re expecting another.
Regarding your license type, we have a great Knowledge Center troubleshooter clarifying if you’ve got yourself a Windows license or a Mac or if you have other problems here: http://help.sketchup.com/en/article/1205653
In the case of this thread, you can tell which type of license you have based on if it ends with W for Windows or ends with m for Mac.
If you’ve ended up with a Windows license and you’re using a Mac, you need to contact the sales team and have them correct it. We do have a charge for changing operating systems if you’re changing platforms later in life, however if it’s an honest mistake within the first few weeks of the initial purchase then we’ll waive the fee and get you up and running but, you still have to reach out to sales.
The important thing for all license problems is to visit the License Troubleshooter first, it will walk you through the different things to be aware of and likely speed up any support requests you need to make.