Go to http://developer.android.com/sdk/index.html . Download and install the Android Studio Package (the whole thing, not the one labeled SDK Tools Only) appropriate to your computer.
Launch the application.
Mac users may encounter the following error:
If so, you should install these two things:
Java for OS X: http://support.apple.com/kb/DL1572?viewlocale=en_US&locale=en_US
Java Development Kit: http://www.oracle.com/technetwork/java/javase/downloads/jdk7-downloads-1880260.html
On first launch, the application puts you into a QuickStart Wizard. At the bottom is an Update hyperlink. Click it to update to the most recent version (currently 1.0.2).
After you're updated, you will be back in the QuickStart Wizard. In order to get past this to the real application, choose the option to make a new Android Project. Pick Android version 2.2 Froyo, and that you want to do a mobile app, but otherwise it shouldn't matter what you pick.
After you get through the wizard, you will get to the real application and are ready to go.