|If the external drive is USB connected......|
USB devices may not work correctly when they're connected to certain USB ports, e.g. ports in a hub or on the front of a desktop case.
See response 3 in this:
External drives require a lot of power, and they won't work properly in all possible USB ports you can plug them into.
See response 1 in this:
Also - examine the USB ports and the USB plugs - make sure the port doesn't have any metal bent out of shape, such that the plug is loose in the port, make sure the plastic blocking plug that prevents you from plugging in a plug upside down is intact, if applicable, and not loose, in both the plug and the port, and make sure the contact fingers in the port are not bent out of position.
If the external enclosure's circuits are being detected correctly, there is an entry in Device Manager for a USB Mass storage device for it under USB controllers. If you have more than one mass storage devices listed there, e.g. other entries can be for a card reader or a flash drive, you can find which device it is in the properties for the mass storage device.
If the external enclosure's circuits are being detected correctly, but there's something seriously wrong with the data on the hard drive or the hard drive itself, the hard drive MAY show up in Disk Mangement and in Device Manager, but may NOT show up in My Compurter or Windows Explorer.
Usually you can't test the hard drive with a diagnostic program when it's in the external enclosure - you have to remove it and install it internally in a computer. If the drive is a 2.5" (laptop) drive, if it's IDE, you can get an inexpensive adapter to adapt it to connecting it to a desktop IDE interface.
Check your hard drive with the manufacturer's diagnostics.
See the latter part of response 1 in this:
(thanks to Dan Penny for this link:)
Hard Drive Diagnostics Tools and Utilities
If you don't have a floppy drive, you can get a CD image diagnostic utility from most hard drive manufacturer's web sites, but obviously you would need to make a burned CD, preferably a CD-R for best compatibilty, on another computer if you need to.
If the hard drive itself tests okay, any data problems found can be fixed, one way or another.