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What does the "insufficient memory" error mean?

The 'insufficient memory' error often occurs in connection with data sync or app installs when your device is running low on internal storage (i.e. phone, not the internal SD card).

Are there any first-aid hints I can follow to solve the issue?

First thing you should check when encountering this problem is how much internal storage is left on your device (space on SD cards, internal or external, is completely irrelevant to this problem). As a general thumb rule, you need at least 25 MB free (reference missing, but this value is frequently mentioned as minimum in this context) what's defined by sys_storage_threshold_max_bytes.¹ SO mentions the "storage low" event being triggered when free space reaches 10%, also see here.

First aid: Go to Settings→Apps→Manage Apps and...

• turn off internet connection (Wi-Fi and/or mobile data) to prevent your device from filling up the space you free with downloads or updates
• delete apps you've installed once and need no longer
• check app details on how much cache they use, and use the "Clear Cache" button to free this storage. This will not delete important data, just temporary cache stuff, so you will not lose anything
• check for pre-installed apps you don't need. Clear their caches and data, then uninstall their updates and finally disable them (so they won't get updated again). While pre-installed apps reside in the /system partition (which you cannot utilize without root), their data and updates consume space on the /data partition which gets freed this way.
• optionally: try moving some apps to other storage (SD card) via App2SD
• On Android devices running version 6.0 and higher there is option called 'Adoptable storage which allows to use SD card as internal storage so you get more space for apps and their data

That didn't solve it. Are there any other recommended sources?

• What can I do to manage my phone's internal storage?
• HTC Desire low on internal storage, I'm clueless — Any help?
• How can I deal with “Low on Space” messages on my Android?
• False “low on space” (internal storage) warning on HTC Incredible
• Why am I constantly running out of space on my Galaxy S2?
• How can I resize partitions?
• also check other heavily frequented questions on this topic

What if I still could not solve my issue?

If your problem does not go away using above "first aid hints", please state so in your question. This avoids unnecessary comments and questions for more details, and we can go straight into other details.

¹ _{in the old days of Eclair/Froyo that was often set to 25MB, today it's usually set higher, e.g. 500M (see e.g. here)}