The reason why I asked as it varies slightly between PC's. I'm assuming your HDD is IDE instead of SATA for this example.
So you will need to use a spare IDE cable, or a socket from that cable and plug it into the IDE connector shown above - left, and also the power connector shown above - right.
If your using the socket on the very end of the ribbon cable, you will to make sure your HDD is set to "Master". If you are using the socket in the middle of the ribbon cable, you will need to set your HDD to "Slave". You can do this by removing the jumper shown above - middle. If you look on the front of your drive, it will tell you which two pins you will need to put the jumper into, to make it "Master" or "Slave".
Once this is done, some systems will pick the drive up automatically. Others you will need to enter BIOS by pressing [DEL] or [F2] when starting, and look for the option to autodetect your HDD.
Once this is done, you will then be able to access the drive from your new PC!
EDIT: Marked as answered I think bacup?