Why is my external drive not being recognized by my computer?

The external drive itself is fine, as well as, my computer. Neither have suffered any damage or have any viruses.
The external drive is pretty much full and I don’t know if that has anything to do with it.
Details:
My external is a Western Digital about 3 years old. It has 160 gigabytes. The model number is WD16001032.
I have Windows Vista Home Premium. My computer is an ACER desktop purchased about 2 years ago.

The thing that I’m having a problem with is that my computer won’t recognize it. It says it’s working properly and that I can safely remove it. But, I cannot get my computer to recognize it. I have an old Dell laptop from 2001 and it recognizes it, but will not open it because it cannot open up a high speed external device. This is a recent occurance and I got some files that I need to get off of it.