Ok try doing this ...
Disconnect ALL the cables from your old computer.
Now re-plugin only the Monitor cable. Try to start your computer and see if its doing the same.
If it is doing the same, shut down computer and plugin Keyboard and mouse cables.
Start your computer again. As soon as it shows ANYTHING on the screen, tap on F8
You will see the Safe Mode Options screen.
Try to start computer with those options. Try every available option one by one.
If this does not work, check in BIOS. You may access BIOS by pressing DEL or F2 or F12. Check if the system is showing the Hard Drive listed. Check it in Boot Options.
If you see nothing wrong in BIOS or dont understand what to see and what not, Find your Operating System CD (Win XP or whatever OS you have on old computer).
Change the boot options in BIOS and make your CD Drive the first bootable device and pop the CD in CD Drive.
Restart computer after saving changes in BIOS.
Press the key when computer says "Press a key to start with CD ROM".
Wait for a minute or two. When you see the options, press R.
Login as administrator. If you provided any password, use it or if not, then leave the password blank and simply hit enter.
Which windows installation you would like to logon to?
Press 1 and hit enter.
You should get C: prompt
Type the following on
C: and it should look like ..
C: chkdsk /r (after typing chkdsk /r hit enter)
This will run a test and the test may take some time unless the hard drive has completely failed.
If Hard drive is 20GB - Test will take 20 minutes.
If Hard drive is 40GB - Test will take 40 minutes.
If Hard drive is 80GB - Test will take 60 minutes.
If Hard drive is 100GB - Test will take 90 minutes.
Be Patient.
Once the test is complete,
on C: prompt, type another command ...
C: fixboot c: (after typing fixboot c: hit enter)
This will take only 10 secs at the most.
Once this is done, type Exit and hit enter
THe computer will restart.
MOST probably, this will work and you will see the desktop.
Otherwise, do what others have suggested, take out the hard drive from your old computer and put it in your new computer.
I hope your old computer and new computer have similar type of Hard Drives. In case if your Old computer has a IDE hard drive and the new one has SATA, God help!
Well, not really, take the old hard drive to best buy or circuit city and get all the data retrieved on CDs.
I hope this helps ... Good Luck!