Actually, look for three more details in the Lower Right pane of Disk Management.
1. For the troubled external drive, does Disk Management show an actual drive with correct size and a valid Partition? Sometimes the unit seems to "appear" here but with no info at all. In those cases, what Disk Management is "seeing" is just the fact that it can communicate with a USB controller in the external case, but it can't actually get useful data from the drive in the case. If that is your situation, the drive MIGHT be all right itself, and just the case is faulty in some way. You can remove the drive from the external case and mount it inside a computer as an internal drive, and see if it works there.
2. If the drive does appear in the external unit with some data correct, What does it say about the File System of the Partition? Normally it should show as "NTFS", although sometimes it might be "FAT32". But if it shows as "RAW", what that really means is that some data bits in its management tables have been corrupted and Windows does not understand it. Then you pursue what alt-rtt suggested - find and use a data recovery program. In addition to what he / she suggested, also check into GetDataBack and Easeus.
3. If the File System does show as NTFS, does the Partition have a proper letter name? Windows cannot use a Partition with no name. So if it looks OK but simply has no letter assigned, you can fix that. RIGHT-click on that Partition and assign a letter to it that is not already in use. When you have, back out of Disk Management and reboot to update the Registry, then see if the drive shows up in My Computer properly.