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Here are the official definitions for these terms
Total user installs - the total number of unique users that have ever installed this app. We only count one install regardless of how many different devices they installed it on. It includes the users who have later uninstalled the app.
Active user installs - the number of users who have the app currently installed on at least one of their devices.
Update: Google has changed these terminologies now.
INSTALLS BY USER - This count increases whenever an user installs the app. This increases even if the user just updated the app to a newer version.
UNINSTALLS BY USER - This count increases whenever an user uninstalls the app. This increases even if the user just updated the app to a newer version.
INSTALLS ON ACTIVE DEVICES - This is typically the difference between 'INSTALLS BY USER' and 'UNINSTALLS BY USER'. As a rare case, if the device on which the app is installed is not active in the last 30 days, then those are excluded from this count.
Original Answer (No more relevant):
I understand this question is old but just wanted to share my experience. I noticed the same behavior as Asterios has pointed out in his comments.
When i checked the app dashboard, i found the following: By the term 'Active Installs', they actually mean Active "Device" Installs. Whereas by the term 'Total Installs', they mean Total "User" Installs.
I think some of us are confusing "Active Device Installs" with the term "Active User Installs". These 2 are totally different.
This page has all the terminology explanations: http://support.google.com/googleplay/android-developer/bin/answer.py?hl=en&answer=139628
That said, I still couldn't make any meaning out of the numbers that I see in my dashboard even though I understand what they are supposed to explain. So, I guess, there is something wrong somewhere in the calculation of those figures.
The Active Install figure can be wrong. I just uploaded the app and installed it on my phone. The next day when I logged in to Android dev backend, I could see 1 Install and 0 Active Install.
I hadn't un-installed the app. This shows that the figure can be misleading.