Desktop hard drives are 3.5" drives. Laptop drives are 2.5" drives. It wouldn't fit.
Old hard drives use the IDE interface. Your new laptop will likely be using the SATA interface.
Sorry to disappoint.
If you would still like to access the data, you could purchase a 3.5" IDE hard drive enclosure that will connect to your laptop via USB.
The hard drive in your desktop will be the big metal and plastic square in a rack to the right of the motherboard when you open the case. It will be about 4" wide. ensure that you have the computer powered off and unplugged before you begin work. Also bee sure to ground yourself on the metal frame of the case to prevent static shock from damaging the parts. Do not work on a carpeted floor. Also do not press on the top of the drive as this will press the head of the drive against the disk.