The first is tablet based, called "Microsoft Surface RT", so to get Overdrive, you treat it like a tablet. Get the Overdrive app and all that, from the Appstore, like you would any Android device or smartphone. Straight from Overdrive:
|Microsoft Surface with Windows RT|
|With the OverDrive app for EPUB & MP3 and the Kindle app for Kindle Books.|
So, you have to either get the Overdrive app or the Kindle app, much like any Android device. I've used both, but I prefer the Overdrive app since you don't have to go through the Amazon website everytime you want to get an e-book. The other great thing is that you can do both audio and e-books from the Overdrive app, so it's one-stop shopping.
Now, back to Overdrive.
Notice how Overdrive doesn't say anything about the other Surface? That's because if you don't have the RT, then you have the other OS. You have the MS 8 based "Microsoft Surface 8 Pro", which is a desktop OS.
To get Overdrive on the Surface 8, you treat it like you would your desktop computer. To do that, you get the Overdrive Media Console for Windows 8, which according to Overdrive, works on any Windows 8 device. To get that, just go here. You just pick your version - which would be the desktop Windows 8.
If you don't want to go through the Overdrive website, you can go right through your Windows Appsore. Do a search for "Overdrive" or "Overdrive Media Console", both of which give you the only Overdrive app in the Windows Appstore, which is the Overdrive Media Console. It's a free app, so download and enjoy!