You should end up with an SDK directory somewhere on your Mac with subdirectories like: extras, platform tools, tools, etc. Nothing can get done until the Android SDK is installed, so let's start there.
Many thanks to James Montemagno for pointing me to Paul Batum's post on the Xamarin forum. If you've been looking for a way to develop your Xamarin Android apps in a Parallels Desktop Windows VM, but run the app in an Android emulator on your Mac, this post is an attempt to distill the information from across multiple sites to get you started.