Apple released an open letter to the public outlining their reasons for not complying with the FBI's demands to modify the iPhone's security mechanism.

Here's a summary:

• The FBI has an iPhone in their possession which they would like to access data from. The phone is locked and fully encrypted.
• After failing to get into the phone, the FBI asked Apple to unlock the phone.
• Apple said since the phone is encrypted, they can't get into it either.
• The FBI asked Apple to modify the iPhone OS to enable brute force password attempts electronically. (Currently the passwords can only be entered in via the manual interface, and is limited to 10 attempts.)
• Apple refused. They believe it would be too dangerous to make that change because in the wrong hands it would undermine the security of all iPhone users, even if they only used the software in this instance.

I understand Apple's position of not wanting to make the change, particularly for new phones, but it's unclear whether the change could actually be made and installed on an existing locked and encrypted phone. Could they actually accomplish this for an existing encrypted phone? If yes, then isn't simply knowing this is possible also undermining the security? It seems to me it would be just one step removed from the backdoor they are trying to keep closed.

Update: since this is a security forum, I feel it is important to point out that Apple is using the word backdoor differently than we typically do on this site. What the FBI has asked Apple to do would not result in a backdoor by the typical definition that we use, which is something akin to a master key. Instead, in this case, if Apple were to comply, the FBI would then be able to attempt to brute force the passcode on the phone. The strength of the passcode would determine whether they are successful in gaining access. Based on Dan Guido's article (linked to in Matthew's answer), if each passcode try takes 80ms, then the time needed to brute force the passcode would take, on average (by my calculations):

• 4 digit numerical passcode: about 7 minutes
• 6 digit numerical passcode: about 11 hours
• 6 character case-sensitive alphanumerical passcode: 72 years
• 10 character case-sensitive alphanumerical passcode: 1 billion years

Obviously if a 4 or 6 digit numerical passcode was used, then the brute force method is basically guaranteed to succeed, which would be similar to a backdoor. But if a hard passcode is used, then the method should probably be called something other than a backdoor since gaining access is not guaranteed, or even likely.

Update 2: Some experts have suggested that it is theoretically possible for the FBI to use special tools to extract the device ID from the phone. Having that plus some determination and it should be possible to brute force the pin of the phone offline without Apple's assistance. Whether this is practically possible without destroying the phone remains to be seen, but it is interesting to note that if it can be done, the numbers I mentioned in the above update become meaningless since offline tools could test passcodes much faster than 80ms per try. I do believe that simply knowing this is possible, or even knowing that Apple can install new firmware to brute force the passcode more slowly, does imply a slightly lessened sense of security for all users. I believe this to be true whether Apple chooses to comply with the order or not.

There are multiple excellent answers here, and it's very difficult to choose which one is best, but alas, there can be only one.