How to debug an Android app's network traffic? I'm trying to listen in on the network traffic of a commercial app I've downloaded. 

Ideally, I'd like the traffic to use my development machine as a proxy, where I'll have Burp or Fiddler waiting to forward requests to the web while I watch the traffic. 

Is this possible with android apps? The app does not have the debugging flag set, of course. I've done no android development, so apologies in advance.
A proxy is a proxy. Unless the app is doing compression or some other protocol like Google Chrome you should be able to observe the HTTP traffic. HTTPS traffic will require certificate for man-in-the-middle to work.

Edit 1:

On the phone/tablet go into Settings->Wi-Fi. Long press on the connection to your Wi-Fi Access point and a menu should popup to modify the connection. Once that screen comes up look for 'Advanced Options', enable and you should now be able to set manual proxy.

When I refer to Chrome I meant their data compression option for Chrome for Mobile:
https://developer.chrome.com/multidevice/data-compression

You'll have to turn this off to sniff traffic from Chrome.