Well, presuming the old drive is IDE, it will be easy. Just crack open the case on the new machine and locate the new hard drive. Follow the ribbon cable back to the MB and you will find the IDE controllers. There should be two of them, each of which will support two IDE devices for a total of four. Depending on what other peripherials you have will depend on how many more devices you could hook up, but you should at least have room for your old HD. On each controller, one device must be set up as master and one as slave. The new HD should be set as the master device on the first IDE controller in the new PC. Depending on how many CD drives you have (i.e., separate CD-ROM and CD-R) will depend on how the rest is set up. If you only have one other IDE device, it could either be set up as the slave on the first controller or the master on the second (though the latter would be the better configuration).
You'll see the ribbon cables will likely each have three females ends on them - one to plug into the controller, and one for each device...
So, what you need to do is examine your existing configuration. If you currently have two IDE devices plugged each, each set as master on a different controller, you'll need to set your old HD up as a slave (as it is likely currently set to master) and connect it to one of the free female ends on the existing ribbon cables. Do do this, you'll need to manipulate the jumpers on the back of the HD. These are just little connectors that bridge two pins on the device. There will likely be three possible positions, one master, one slave, and one that is supposed to autodetect the state, but that is not really recommended. They should be labeled in some way, but it may be on a sticker on another part of the HD and not right next to the jumpers.
If you have two devices and they are currently connected to the same controller on the same cable, then you can keep the old HD set up as master and just connect it to the second controller. You'll likely need to supply your own cable for this.
If you have three devices (like the HD, CD-ROM, and CD-R), just find the controller with a free space, plug your old HD in and make sure one device is set to slave and the other to master.
That should be it hardware-wise. You may have to go into your BIOS and set the state of whatever controller you plugged the old HD in to autodetect - instructions for your BIOS should come with the new PC. Then just fire her up and the old device will appear as a new drive letter. You can copy the files you need over, then reformat the old HD and keep it in there for some extra storage space. If it is an older PC, it is likely a slower drive than your existing one, so I wouldn't install any programs on it, but it would be a good place to keep mp3's, pictures and such if you are into that.
Rocko