In an official statement on Tuesday, Apple states that the attack was not any sort of security issue, but rather a targeted attack using usernames, passwords, and security questions to gain access. Apple states that this sort of attack "has become all too common on the Internet".
Via Apple:
We wanted to provide an update to our investigation into the theft of photos of certain celebrities. When we learned of the theft, we were outraged and immediately mobilized Apple’s engineers to discover the source. Our customers’ privacy and security are of utmost importance to us. After more than 40 hours of investigation, we have discovered that certain celebrity accounts were compromised by a very targeted attack on user names, passwords and security questions, a practice that has become all too common on the Internet. None of the cases we have investigated has resulted from any breach in any of Apple’s systems including iCloud® or Find my iPhone. We are continuing to work with law enforcement to help identify the criminals involved.
To protect against this type of attack, we advise all users to always use a strong password and enable two-step verification. Both of these are addressed on our website at http://support.apple.com/kb/ht4232.
Apple is currently working with law enforcement to help identify who the attacker(s) was who leaked the photos. The anonymous 4Chan user claims to have more of these photos and is now asking for donations in Bitcoin and other currencies to release them. Although Apple said security in iCloud was not an issue here, the company did claim that a recently-patched Find My iPhone bug could have played a part.