One method:
Open up the PC case. (Dead easy with most makes but possibly a real pain in the backside if it's a Dell!). Unplug and remove the hard drive. (There will only be a couple of screws holding it in). Read the label to see whether it's an IDE or SATA drive. (Alternatively just examine the connections. The top one here is IDE; the bottom one is SATA:
http://gallery.techarena.in/data/513/sata-ide_lg.jpg )
Purchase a 3.5" USB enclosure, making sure that it's the right type. (i.e. IDE or SATA. Some enclosures can handle both types. e.g.
http://www.cclonline.com/product/76329/U35SPA/External-HDD-Enclosure/CiT-3-5-USB-2-0-SATA-IDE-HDD-Enclosure/HDD3443/ )
Fit your old hard drive into the enclosure. (Easy peasy).
You've now got an external USB drive that you can plug into your new PC's port. It will show as an external drive but, initially, your data won't be visible. That's because you need to take 'ownership' of the drive, which is explained here:
http://www.preyerplanning.com/take-ownership-of-entire-hard-drive-in-windows-7.pdf