The government is asking Apple to hack our own users...
[T]his demand would undermine the very freedoms and liberty our government is meant to protect.
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Tim Cook, Apple CEO
Apple refused the order by the government to create a "backdoor", claiming that such a device would make it possible to unlock any phone. This could be used by any to access anyone's private information.
Tim Cook in an appeal to it's users wrote an open letter, arguing this precedent could lead the government to access "...financial data, track your location, or even access your phone's microphone or camera without your knowledge".
[1]
What Happened
On December 2nd, 2015, two Americans attacked a Department of Public Health party, killing 14 and injuring 22 people. This led to an investigation led by the FBI.
However, the FBI wasn't able to decrypt one of the terrorists' phones, and in trying to access information, ordered Apple to provide a way to unlock the phone. Eventually, the FBI resolved this by finding a way to access the information, but this question of responsibility is unresolved.
[2]
The government argues that Apple is avoiding helping the government access key information to protect itself. In a statement, the government said "Apple's rhetoric is not only false, but also corrosive of the very institutions that are best able to safeguard our liberty and our rights: the courts, the Fourth Amendment, longstanding precedent and venerable laws, and the democratically elected branches of government".
[3]