The Disappearing Drive
The first step towards recovering lost data is figuring out why the hard drive is no longer working. There are many reasons a hard drive could disappear or become non-responsive, and most of them don't mean that the hard drive itself has failed. A computer, with its army of drivers, can be finicky to changes.
In order to determine if your hard drive has failed, you'll want to test the hard drive in another computer. This means taking the hard drive out of your current PC and placing it into another one. This is easier than you might think, because hard drives are plug-and-play devices, so long as you're dealing with a modern SATA drive. Once you have installed the hard drive into the new machine, Windows should detect it and make it visible. If you're moving a drive between PCs that are using different versions of Windows, then the test PC may very well ask to reformat or modify data on the hard drive you're trying to save. Be sure to pay attention to all prompts, so that you do not accidentally reformat the drive you're trying to retrieve data from.
Of course, it is likely that you don't have a second computer on which to test a hard drive. If that is the case, then you'll want to buy a SATA to USB adapter. These can be found at your local Radioshack or local PC repair shop, although you'll save a lot of money by buying one online. Once you have the adapter, use it to plug your hard drive into a USB port on your PC. Doing this should cause the hard drive to appear so long as it is still alive.
If either method causes your hard drive to appear, go ahead and grab your most important files before placing the hard drive back into your PC to search for the cause of the drive's disappearance.