Feel free to add your tips here!
PREFACE: OK - you should always back up any important data. And then back that up and keep the second backup somewhere else, so even if disaster strikes your computer and the on-site backup the off-site backup will probably remain intact. If it's over 5 miles away, and both get wiped out, you've probably got bigger problems than loss of a couple of computer files to worry about. And if the backups are current make sure they're readable. A backup of a corrupted disk isn't all that much help. But if you have verified backups, you won't have a whole lot to worry about.
So, you have lost some files, card has died or a disk does not respond.
The first thing to do is remember the Hitchhiker's Guide to the Galaxy and those large, friendly words on the cover:
1. Prevent any more damage
Before you do anything else, stop using the disk/flash card/USB drive/papyrus sheet/clay tablet they were stored on. If this was the main drive of your computer, shut it down promptly.
The more opportunity you give your computer/camera/MP3 player to write new information to your storage device the greater the likelihood you'll overwrite some of the information you want to recover.
If the drive was your computer's operating system drive, take it out from the computer. Kids next door know how to do it, don't be afraid to ask!
If the drive was a system drive, you will usually need another computer for recovering the files.
2. What is the specific problem?
Operating system loads, but files are not there where they should be.
Perhaps you forgot to power on an external drive, or connect a cable. If devices show ok, do a search! Check the trash can. Don't panic, think.
I deleted some files and folders and want the back.
On Windows you can try first pressing CTRL+Z. This is a standard UNDO command which works on file operations, too, only if you do it right after you deleted files.
If undo does not work, make sure the disk is not accessed, then download an undelete program, see http://photography-on-the.net/forum/showthread.php?t=670801
Computer does not respond, reboot does not help.
Does the computer make any sound on boot? If not then most likely power supply has died.
Do you get any blue screens or BIOS (boot screen) error messages? This usually means there is a bad driver or hardware error, good news is that you data is still there. Google the error message for more into what might be the problem.
If computer boots and gets "stuck", it might suggest drive failure. Do you hear hard disk spinning on powerup?
Camera card says it needs to be formatted.
File system might be incompatible with your OS. File recovery programs can read most filesystems. See http://photography-on-the.net/forum/showthread.php?t=670801
I accidentally formatted the card / disk drive.
If you did Quick Format, you can most likely recover all data. Get a recovery tool from http://photography-on-the.net/forum/showthread.php?t=670801
If you did low level format, recovery needs to be done by a recovery company.
I dropped a laptop/storage device.
Physical damage to a disk will be impossible to repair at home. Recovery needs to be done by a recovery company.
Home made data recovery in detail
Before you start trying to recover the files, make a "bit-copy" of the drive. Some of those utilities will help you do this. There are also specific "drive cloning" applications that can do this. Again, try before you need them for real. And make your first rescue attempt(s) on that clone. This way, you won't ruin your original if something goes wrong.
Next (if you haven't gotten one or more already) download some of the applications from here. In fact, go out and download a couple of them now, before you need them, so you can practice recovering files when there's no stress. I'll wait.
If your computer won't recognize your disk or card, you'll need to do a bit more work. First, if you have another device that's supposedly able to read it, try reading from that. A computer that uses a different operating system (Windows or Linux if you're a Mac user, Linux for the Windows user, Windows for the Linux user) is ideal, but you'll need to know at least a little about that other OS. You can always download a bootable version of Linux that'll run from a DVD or USB drive and see if that helps. If nothing else, the "dd" command can often make an exact copy of the original, failed media. Or if it's your camera's memory card, see if you can download direct from the card.
If the hard disk doesn't want to respond, get an external drive housing and mount it in that. If your computer has room for another internal hard disk, and an external drive has failed, try putting the drive into one of your computer's hard drive bays. Sometimes it isn't the drive, but the supporting electronics, that failed. For the tech-savvy (or daring), if you have a second, identical, hard disk, you might try swapping the external circuit board that's part of your hard drive for the one on the second drive (but make sure you back up any data on the second drive at least twice first).
Once you've made a copy of the problem drive, try one of the recovery utilities you previously acquired. It may need to run a while, especially if your drive is really corrupted, and/or you're trying to recover from an accidental reformatting or mass deletion of files. And remember, these utilities aren't always at their best with proprietary file types, like RAW files. They may recognize the embedded .JPG files and give you huge files containing a low resolution JPEG, and with a .JPG extension. So you may have to go back and rename your RAW files and see if that helps you read them.