|"Most probably it is mapping to a drive letter that is already in use...."|
(Logical) drive letters are assigned to USB drives according to what drive letters are available in the operating system when the drive was plugged in - there should be no problem regarding mapping, unless all the drive letters C to Z are being used when it was plugged in.
Your external hard drive is apparently 3.5" and in that case it came with an external power adapter that you plug into it - that MUST be plugged into it, and working.
If the USB cable you are using between the external drive and the computer did NOT come with the drive, it may be inadequate and you may need to try another one. The cable must be rated for USB 2.0 use, and must be able to handle the full max 500ma USB spec.
For a desktop computer, you may have an IRQ sharing problem you need to fix. In that case, some USB devices work correctly, some don't.
See response 3 in this: