In one of the tek episodes Wendell said that he wanted to learn how android works. I was going to make my own list of tools for exploring android and email it to him. But then I decided that a forum could make it better and it could be a public list.
Please add to this.
<a href="https://play.google.com/store/apps/details?id=stericson.busybox&hl=en">Busybox (root, required)</a>
This is a collection of common unix utilities.
<a href="https://play.google.com/store/apps/details?id=jackpal.androidterm">Terminal Emulator (root, optional)</a>
This gives you a CLI shell on your android computer.
<a href="https://play.google.com/store/apps/details?id=eu.chainfire.supersu">Super su (root, required)</a>
This lets you manage which apps are allowed to use the root account.
<a href="https://developer.android.com/sdk/">Android sdk</a>
This is a set of developer tools that run on a separate machine for managing an android computer.