PCSX2 is a free PlayStation 2 emulator that has been in constant development for nearly a decade. It is plugin-based, providing multiple options for emulating the CPU, graphics card and other components of the PlayStation 2 as fast and as efficiently as is possible. Improvements to the software over the years and better computers have made it possible for PCSX2 to often run games at their native frames per second, with no lag.
It's simple enough to download and install, but before you can use it, you need to obtain a PS2 BIOS, which can be either simple or difficult, and you also need to select and install plugins for each component of the PlayStation 2 that you're emulating. This isn't as hard as it might sound, and the software's website has several useful guides for doing this, but it's still a barrier to entry when it comes to emulating PS2 games. Once you're playing the games though, PCSX2 is pretty much perfect: fast and accurate emulation, just enough control to make sure everything works as well as it can, gamepad support, and even debugging tools if you need them.
The only weakness PCSX2 has is that, even with improvements to the software, machines from more than a few years ago will struggle to emulate PS2 games that well. If your computer is reasonably modern, and you're willing to configure it, this software will work for you.