If you're lucky, maybe you just don't have the jumpers set correctly and your BIOS didn't pick the drive up.
Check to see what jumper your HDD that is already installed in the new PC is set at. You should see something like MA, SL, or CS. If it's on MA, you need to put your crashed drive as SL. If it's CS, put your crashed drive as CS.
MA = Master
SL = Slave
CS = Cable Select
Now that you have the jumpers correct, in your BIOS, you might/should have an IDE verification section. If not, look for something like "Standard CMOS Setup". You should see a list of Primary Master and such. Just put them all as "Auto". If the jumpers are set correctly, keep your fingers crossed that it can still be recognized.
I'm looking for a guide that shows you the basics of your drive.
EDIT -
this is from Seagate, but principles are pretty much the same across the board.