RPCS3 is an open-source Sony PlayStation 3 emulator and debugger. It has been in development since 2011. It can boot and play a decent number of commercial games. Like most emulators for recent consoles, you will need a decent gaming PC though.
You can find a quick start guide on how to get RPCS3 running on the official site. You will need two things to run the emulator: the Microsoft Visual C++ 2015 redistributable and the PS3 firmware. You can download the firmware straight from the official PlayStation website.
You will also need games of course. It isn't that hard to make a backup copy of a real PS3 game disc, but it's a bit more complicated than copy-and-paste. The detailed instructions can be found in the quick start guide. First, need to create an ISO file from the PS3 game disc using an application like Alcohol 120%. You will also need a compatible Blu-ray drive for this. You then need to patch and decrypt the ISO file. Finally, you need to extract it (put the contents in a folder with an appropriate name).
At the time of writing (spring 2017) I'm not sure if I would consider RPCS3 to be at a playable level. It is not yet at a point where it could serve as a replacement for a real PS3 or the PlayStation Now service. But it is exciting to see how far PS3 emulation has come.