I advise you backup all your data immediatly. Also, if you're using Vista I used to have an issue where if after I had selected to move files, if I did anything else the file movement would halt, however if I just left the the computer alone it would move fine. That issue is now gone for me but it's food for thought.
How can I make a backup if the hard drive switches off when I am moving around files?
It's not a good thing. But it's not necessarily a bad drive.
First, does it do the same thing on another computer? If not, then the problem's possibly with your USB or disk drivers. See if it works with a USB flash drive or another external hard drive.
Is this an external HDD with USB power, or does it have its own power supply? If it's USB-powered, your computer may not be sourcing enough current. USB-powered drives usually come with dual USB connectors, and they work best if you connect both. If it has its own power supply, the power supply may be going bad. Depending on the power supply, it may be easy or hard to replace.
Can you read (as opposed to copy) files on the drive? What noise does it make while it's reading (or trying)? A drive that's going bad will produce a steady series of clicks or clunks at 1 or more per second, as it tries to reset and re-read.
If this is a take-apart housing, can you take out the drive and try it in a different housing, or in your computer? There's a controller in the housing that can itself go bad. (For this reason, I don't buy proprietary preassembled external drives; I buy a drive and a housing and assemble it myself. That way I can always take it apart or reuse the drive or housing if the other part fails.)
I don't currently have any other computer to try it on. But, this has never happened before.
It's an external HDD which is USB powered. It is several years old, if it matters. As for the reading, sometimes when I open up a folder, the power light blinks and makes makes about 1 or 2 sort of "scratching" sounds(which I assume is the steady series of clicks you are talking about).
What is failing? The internal HDD or the external one?
External.
Edit: Well, I was able to make some progress. If I move a folder or files that equal less than 30MB in size, it will transfer okay. Anything, larger, and the hard drive just switches off. I have about 20GB of files in the HDD, so it may take a while making backups of everything 30MB at a time.