Amazon is one-upping Apple.

The retail giant has launched a streaming service, allowing users to store their digital music (and other files) online and play them anywhere via the Web or an Android smartphone or tablet.

Dubbed Cloud Drive, the service offers 5 GB of free storage to all users. That space is capable of holding up to 1,000 songs, 2,000 digital pictures or 20 minutes of HD video. For people with larger collections, the company is offering a range of tiered storage plans costing anywhere from $20 to $1,000 per year. (Amazon is also offering a free one-year upgrade to a 20 GB package to customers who buy an MP3 album.)

Music is the initial focus for the service — a natural move, given the long-standing rumors that Apple is thinking about moving its iTunes service to the cloud, a technical term for online content storage that is streamed to computer and mobile devices via the Internet. Amazon is adding a “Save to Amazon Cloud Drive” for all digital music purchases to encourage use of theservice and is offering an easy (if currently quite slow) option to let users upload songs from their computer’s hard drive.

Video, photos and documents can also easily be stored and uploaded in the account, but any files with DRM-protection are not allowed (something that isn’t likely to slow down owners of pirated content). Users will be able to access their files via most Web browsers or on their Android device via a proprietary player from Amazon.

The files cannot be accessed via the iPhone, however, as Amazon has seemingly blocked streaming through the Safari browser.

The launch of Cloud Drive gives Amazon bragging rights in the high-tech world — as both Google and Apple are working on similar cloud services, which will let consumers access their content when they’re away from their primary computer. Apple has an enormous data farm in North Carolina that is expected to be the hub of its cloud operations.

More importantly, it gives Amazon the chance to establish a customer base before the other services launch (likely in the next three to four months). Cloud storage sites traditionally have been like email accounts for users — once they have taken the time to set up the service and use it a few times (or, in this case, have loaded their files to one), they quickly become embedded and are less likely jump to a competing one.