Mobile App Development
There are two main mobile operating systems: iOS from Apple and Android from Google. Both mobile operating systems support two main mobile devices or form factors: the mobile phone and the tablet. The devices are defined primarily by their screen sizes: phone screen 3.5" - 4.5" and tablet screen 7" - 10." The app development environments below will be organized primarily by operating system with the caveat that some development environments can create BOTH iOS and Android apps.
iOS App Development
xCode (free from Apple). More mobile apps have been developed in xCode than any other. It is provided free for Mac OS X computers as part of their Developer Tools. Visit the Apple Developer Website and download xCode from the iOS Dev Center: http://developer.apple.com. The iOS Dev Center has training materials and sample code. Another great resource is the iOS development course from Stanford University you can find on iTunesU.
Android App Development
AppInventor (free from MIT). http://appinventor.mit.edu