App Inventor Classic is an older version of App Inventor. App Inventor 2 is the current version. App Inventor is a free system originally from Google for creating applications for mobile devices that use the Android operating system. App Inventor moved to MIT in 2012. Go to http://beta.appinventor.mit.edu/ to use the classic version of App Inventor. For information on the classic version of App Inventor see http://appinventor.mit.edu/explore/classic.html
To get a feel for the types of things you can create with App Inventor see http://gallery.appinventor.mit.edu/#.
In App Inventor you create the user interface and select components for your mobile application in a browser using the Designer and then program it with a drag-and-drop Blocks Editor than runs on your computer using Java Web Start. App Inventor comes with an emulator so you don't even have to have an Android device to test your application. But, if you have an Android device you can run your apps on it. However, installing device drivers can take some time. You can also download to your phone over a wireless network without installing device drivers.
You will need a free Google account. You also need to install Java and the Android SDK on your computer to use App Inventor. On windows computers if you want to install the apps on your phone you need to install a device driver for your phone.
You can upload your app to the Google Play! website - see the following for directions (there is a registration fee of $25 to register as a developer) http://beta.appinventor.mit.edu/learn/reference/other/appstoplay.html
For more information see What is App Inventor.
To set-up your computer and phone for App Inventor see Set-up for App Inventor.
For the device driver for the Droid X click here
For instructions on how to manually install device drivers on windows see http://developer.android.com/sdk/win-usb.html#InstallingDriver.
Once you finish setting up your computer and phone you can do the tutorials.
To create a project go to http://beta.appinventor.mit.edu/.
When you want more information on the user interface components see the Reference Documentation
Below you will see links to documents that include a powerpoint introduction to App Inventor, an App Inventor reference, and an user interface reference.