This is a little bit thin to work with. What make and model are the two computers? Are the hdds both IDE/ATA, or are they SATA hdds.
If they are both IDE hdds and you are using a PC install the hdd from the non-functioning computer as a slave drive on the same IDE cable that the other hdd is connected to as the master. You will have to set the jumper on the rear of the two hdds so the the one connected to the end of the cable is set as Master (M), and the Slave hdd is connected to the middle connector and set the jumper as Slave (S). There is a good tutorial for this installation, you can read it here
If this is a laptop you will need an external enclosure that connect via a USB port.
Once you have the hdds go to My Computer and see what the new hdd is being recognized as under Hard Disk Drives. If this is connected to a USB port it may show up as a removable storage.
Right click on start, click on Explore. Go down the list on the left side till you find your old hdd, double click on it and the contents should appear on the right side of the screen. You can click on the file that you wish to move and hold the button down and drag this over to your C drive, the C drive will highlight in blue, release the button and it will have been moved to the C drive.