I have an old PlayStation 1, and I am a big fan of old games. I have a lot of game CDs for this PlayStation.
Is it possible to run these games on my PC? If so, how?
Yes, you can run them in an emulator such as ePSXe. Typically emulators require rather powerful machines to run at the kind of FPS you'd expect from the console, however you're unlikely to experience issues with a Playstation emulator and a half-decent PC.
For more information on Playstation emulation, plus a list of emulators, check out the Emulation General Wiki.
Just keep in mind that, in most cases, downloading the games themselves is illegal.
take one of the emulators from this list, install it and play away. You might want to invest in a gamepad as it will feel a lot more like playing on the actual ps1. If you have a xbox controller, it should be plug and play with the first one on the list, if i remember correctly.