Update: The Beta version of Cortana just released for the Android devices, and all you have to do to get this Microsoft personal assistant on your devices is to become a beta tester.
To become a Cortana Beta tester on Android, do as follows:
1. Go to Cortana Beta Testers page and click the "become a beta tester" button
2. Now, download and install the Cortana app from the Play Store (Click to download)
3. Open the Cortana and enjoy it on your Android device
** Note that, it may take a few minutes between becoming a beta tester and being able to download the app from Google Play.
** Also note that the beta version of Cortana may not be available in your country yet.
Cortana is the Microsoft's late answer to Siri, Google Now and recently Hound as a premier intelligent assistant. While the official version of Cortana for Android is still on the way, you can do as follows to get the Microsoft Cortana on Android devices:
1. Go to Setting >> Security and make sure the Unknown Sources option is marked (This option allows the device to install application from sources other than the Google Play Store)
2. Download the Cortana APK File from here (Courtesy of Android Police)
3. Install the application and enjoy the fully functional Cortana on your Android device
** Note that this is the unofficial and beta version of the Microsoft Cortana and you may not experience the best functionality)