How can I recover my external hard drive data?

Logical failure/accidental format/undelete . When connecting the drive, you hear that disk spins up, but you see some error messages, or you cannot find the needed files, or the disk not showing up in Windows. If you can't see the disk in Windows, you need to look for it in Disk Management (click Start, type diskmgmt.msc in the search box and press Enter). If you deal with a logical failure, you should see the drive in there.
Disk enclosure damage. When you connect the drive to the PC, nothing happens. The light on the disk doesn't blink, the disk doesn't spin up.
Physical disk damage. When you connect the drive, you hear a clicking sound or permanent disk spinning, or do not hear disk sound from the disk.

Logical failure/accidental format/undelete. To recover data from the hard drive, you have to use a special data recovery software like ReclaiMe. This software is highly user-friendly, so you shouldn't face any trouble using it. Everything you need to do is to download, set it up, run, select your hard drive and click Start. You can find more information about the recovery process on this page - External drive recovery tutorial.
Disk enclosure damage. You will have to remove the drive enclosure, and connect the disk to the PC either using a SATA cable or another enclosure. Different drives are disassembled in different ways. For more deatils, refer to this site which contains the most popular external drives disassembling videos: www.external-drive-recovery.com.
Physical disk damage. There are almost no DIY approaches to recover data from a physically damaged drive. The only thing you can do, given that the drive hasn't died yet, is to create a disk image file and try to recover data from it. Read more on how to create a disk image here. However, if the data is really important for you, it is better to send the drive to a data recovery service where specialists can replace damaged components of the drive.