You can share your app in an executable form (.apk) that can be installed on a device, or in source code form (.aia) that can be loaded into App Inventor and remixed. You can also distribute your app on the Google Play Store
Sharing your app so that others can remix (.aia file)
Make sure you are viewing the list of all of your projects (if you are not, choose Projects | My Projects). Select the project you wish to share by checking the box next to it. Choose Project | Export selected project (.aia) to my computer to export the source code (blocks) for your project. The source code is downloaded in a .aia file.
If you send it to a friend, they can open it with Project | Import project (.aia) from my computer.
Sharing your app for others to install on their phone/tablet (.apk file)
Package the app (.apk file) by going to the "Build" menu on the App Inventor toolbar.
Select "App (save .apk to my computer)." A pop-up box should alert you that your download has begun. Note: The other option (provide QR code for .apk) produces a scannable QR code that will download the app for two hours. You can share this code with others, but they have to use it within 2 hours of your generating it.
Once the build completes, you can email the app (".apk" file) to your friends who can install it by opening the email from their phone. If you want to distribute it more widely, you can upload it to a website that both you and your friend can access. You can also distribute your app on the Google Play Store.
NOTE: Anyone installing your app (which is an ".apk" file) will need to change the setting on their phone to allow installation of non-market applications:
To find this setting on versions of Android prior to 4.0, go to "Settings > Applications" and then check the box next to "Unknown Sources". For devices running Android 4.0 or above, go to "Settings > Security" or "Settings > Security & Screen Lock" and then check the box next to "Unknown Sources" and confirm your choice.
Note: The source code (.aia) files are not executable Android programs -- those are .apk files. The source code is also not Java SDK code -- it can only be loaded into App Inventor.