We need the exact error message (or suitably quoted chunks of it) to be able to help. The colour of the screen helps (is it black, blue or something else?), and an idea of what else is happening - for instance is the computer taking longer to boot than usual? What changed? Is this sudden, or has it been showing similar symptoms for a while? And are there any strange noises, clicks, whirrs or beeps going on? Oh, and the Operating System helps as well!
After all that, I'd put money on it being a dicky hard drive.
If so, here are the basics:
The best thing to do with a failing hard drive is to cut and run - turn the computer off for now and don't try and boot it again (don't worry there is hope).
The Operating System (Windows, Linux, whatever) will not be rescuable, the important thing is to get off as much personal data as possible.
Get the owner to sit down and think about what they want off the disk - is it photos, documents, email, recipes, software licences, plans for conquering the world etc. Once they know what they want back, they need to prioritise - the rescue process is riskier the longer it takes. Some data may not be rescuable if they are using encryption, but I'm ignoring that for now.
Get whomever to arrange for the following:
1) Pull the hard drive that's at fault from the dodgy machine
2) Acquire an external hard drive caddy that is either USB2 or Firewire. Make sure that it is externally powered, and install the disk in the caddy.
3) Find a PC that is capable of reading that disk (Windows 2000 or XP for Microsoft, any recent Lunix box for most other OSs, and I'll let the Apple crowd chip in details for MACs). That PC needs to have enough storage for everything that needs to be rescued.
4) Boot the PC, take a deep breath, think about what you're doing, and plug in the dodgy disk's caddy.
5) Working quickly and methodically, ignore any requests to fix the ailing filesystem and copy off everything that you can, in the agreed order.
While working, look around the filesystem as there will inevitably be files that hadn't been thought about.
It's likely that a new hard drive in the ailing PC will be enough to fix it - it'll be a full reinstallation for that. I suggest that you then copy back the rescued files (there are various methods here, but you can take your time now - I suggest burning them onto DVD if possible). The ailing hard disk can then be plugged back in if necessary to rescue any further files, if it's still working at this point.
I suggest that the owner then buys another new harddisk and uses it in the caddy for backups in future
