The only way to have albums is by syncing folders on your PC to your iPhone with iTunes. These albums will only have the photos that you had in those folders on your PC. In order to add photos to the albums, you must first add them to the folder in you PC, then sync the iPhone to them. So moving a photo from your camera roll to another album within the iPhone is not an option.
It's easy enough with a PC to move photos around though. The iPhone is treated like any other drive. So you could just drag the photo you want from the iPhone into whichever folder you want, then sync with iTunes. But you HAVE to use your PC in order to move photos around.