Anki is a program which makes remembering things easy. Because it is a lot more efficient than traditional study methods, you can either greatly decrease your time spent studying, or greatly increase the amount you learn. AnkiDroid is the Android port of Anki, and is compatible with Anki data.
Opt-in synchronization uses the non-free AnkiWeb service by default, but this can be changed in the settings to use, for example, an instance of the unofficial Anki Sync Server).
Opt-in / off-by-default crash reporting will send data to a private / AnkiDroid open source team controlled crash reporting server if enabled. This data is only used to help fix crash bugs.
Opt-in / off-by-default analytics will send data to Google Analytics via an an open-source implementation of the analytics API if enabled. This data is only used to focus developer efforts on popular features.