Internally, WordPress organizes all media you upload chronologically into directories based on the month and year. I doubt that is going to change anytime soon. However, the main way that the user is meant to organize photos is by attaching them to specific posts and pages.
Currently, you do this by uploading those media items while editing the specific post or page. When you upload a bunch of images when editing a page, for example, those images then become “Attached” to that page and are now makeup the “gallery” of that page. But after you’ve uploaded the images there’s really no way to modify how they’re attached to which pages without deleting and re-uploading (unless you find a 3rd party plugin). This is pretty much the extent of WordPress offers in this regard.
However, the Media Manager is the big focus of the upcoming WordPress 3.5 release. They haven’t really changed how the overall system works, but they’ve now made it so you can manage these “Galleries” much more efficiently, and I believe that is how you’ll want to start managing your media.