I'm one of the founders of Bugsnag which we designed for exactly this purpose. Bugsnag automatically captures unhandled exceptions in Android apps and sends them to our dashboard, where you can prioritize fixes and dive into diagnostic information.
Here are some important things to consider when selecting or building a crash reporting system, along with some code snippets:
If you want to see some best practices around crash handling/reporting on Android you can check out the full source code for Bugsnag's crash reporting library which is fully open source, feel free to tear this apart and use it in your own applications!