Your phone is running out of space, you are moving to a new device, or you simply want years of camera roll backed up somewhere safe. The job comes down to one question: which phone, which computer, cable or wireless. Each of those four combinations has its own trap; on Android the computer sees nothing until you change the phone's USB setting, and on iPhone the full-size version of a photo may not even be on the phone.
The steps below are split by route, so reading the one that matches your setup is enough. Everyone should read the last step, though, because assuming a transfer finished and then deleting from the phone is the most common way people lose photos.