You can play both Playstation and Playstation 2 games on PC using emulator softwares.We don't support piracy here so do this only if you have a original Playstation disc.And you will need to dump your bios from your own PS2,since the emulator needs those files to play the games on your pc.For further details on how to configure the emulators you can checkout the forums and official websites of the emulators themselves.
Here is the link for PS2 emulator http://pcsx2.net/downloads.php
download the first one named
pcsx2 0.9.6 installer.And here is its configuration guide http://forums.pcsx2.net/Thread-Official-English-PCSX2-configuration-guide-v0-9-6
That was for Playstation 2,if you want to play only PlayStation 1 games then download epsxe from this page along with the config utility http://www.emulator-zone.com/doc.php/psx/epsxe.html
Thanks for using Fixya!