There is something called the logcat, which is a combined message pipe from all applications. To read it you first need to locate the adb tool that comes with the Android SDK. Depending on which version you have of the SDK the adb is located either under <sdk>/tools or<sdk>/platform-tools.
Simply start it like this:
$ adb logcat
and it will start printing out everything that is going on on the device. To limit it to only show the output from inside Unity, you can try this:
$ adb logcat -s Unity
or, to get a little bit more info about what’s going on:
$ adb logcat -s Unity ActivityManager PackageManager dalvikvm DEBUG
If you wish to report a bug or otherwise ‘quote’ the logcat, you can dump the complete logcat to a file like this:
$ adb logcat -d > logcat.txt