Don't despair - there is an easy answer, although it'll cost you about $15 if you don't have the extra piece you need (or can't borrow it). Here we go:
0. Every modern operating system boots in the same basic way: Mac, PC, Linux, etc. The operating system kernel (the base layer) loads a couple of critical drivers, one of which is a driver for the hard-drive controller interface (SATA, SAS, whatever). When you move a hard drive to a new computer, there's a good chance that the driver needed isn't present, so Windows fails to boot on an "INACCESSIBLE_BOOT_DEVICE" error, and typically reboots. In Linux, you get a "kernel panic, unable to mount root filesystem", and on a Mac you see what Apple wants you to see (which isn't much use). Solution: get the necessary driver in place -before- moving the hard drive to the new system.
1. Get your hands on an "offboard disk controller" card, where in your case it's probably a SATA controller. You want one that can be plugged into both systems, old and new, so if the new system has no older PCI slots (only PCI-E) then the disk controller must be a PCI-E slot device. This item can be acquired from many sources, like amazon.com, BestBuy, etc. Get el cheapo; it's just a tool.
2. Install the disk controller card in a slot in the old system. Boot up, and feed it the driver when Windows detects new hardware. (The driver should be on a CD that came with the card, if it's needed.) Make sure the new disk controller is working properly (check Device Manager under "Storage Controllers" to make sure there are no unidentified devices). Shut down, unplug the hard drive SATA cable from the motherboard, and plug it into a port on the SATA card. Boot back up. We need the system to boot with the hard drive attached to the card, which it should.
3. Assuming you booted from the drive attached to the SATA card, now shut down and remove both the hard drive and the SATA card. Install both in the new system, with the hard drive cable plugged into the SATA card as before (not the motherboard). Power up the new system - the system should boot, although it'll ask for all kinds of new hardware drivers. Cancel/skip the driver requests for now. Feed it drivers later - this is another couple of paragraphs in itself, which I'll skip because it's really a different topic.
4. At this point, if all went well, you've booted the old drive on the new system. Shut down again (sorry), remove the SATA controller card, and plug the hard drive cable into a SATA port on the motherboard. Boot up again. If you get here, you're pretty much done, except to the part about "loading a bunch of new drivers".
** This can be done without the SATA card, but it requires editing the registry manually, and if you do it wrong (which is easy), the drive will stop booting ALTOGETHER, and I'd hate to see that happen on my advice. The risk involved in this hardware shuffle is much less. "Can't boot" registry SNAFUs can be fixed, of course, but it's an even more ticklish process, so don't go there unless you're ready.