It was back in September of 2011 that Apple filed suit against Amazon for violating the "app store" trademark. Apple applied for this trademark but has not yet been award the trademark.
Amazon requested and argued in court that the "app store" term has become generic. Amazon says that "To the contrary, the use of the term 'app store' to refer to stores selling apps is commonplace in the industry. "Both Steve Jobs (co-founder of Apple) and Tim Cook (current CEO of Apple) have been trying to compete with different storefronts for using "app store."
Other major technology companies like Microsoft are trying to block Apple's application with the U.S. Patent and Trademark Office,
According to Bloomberg, U.S. District Judge Phyllis Hamilton, granted Amazons request, and says that she found "no support for the proposition that Amazon has expressly or impliedly communicated that its Appstore for Android possesses the characteristics and qualities that the public has come to expect from the Apple APP Store and/or Apple products.”
Apple continues to believe that its iOS platform made the word "app" popularized and that most customers associate the words "app store" with the iOS App Store.
The case is Apple Inc v. Amazon.com Inc et al, U.S. District Court, Northern District of California, No. 11-01327.