EmuNations

PCSX2 is an open-source emulator using a combination of MIPS CPU Interpreters, Recompilers and a Virtual Machine through plugins to manage hardware states and PS2 system memory. It has been running for more than 10 years and can run over 2000 titles with many at full speed. 64-bit versions are being developed and tested, please check here.

PCSX2-EX

• It's an extended version of PCSX2 focusing on reversing and modding applications.

Spectabis 

• It's a custom frontend for PCSX2 which removes the tedious task of tinkering the settings for each game or creating complicated setups for each. 

Winsock based DEV9 plugin 

• This plugin uses sockets instead of Tap or WinPcap, which the author hopes will eliminate most configuration issues users face when setting up.

It's an experimental high-level emulator PS2 emulator with the goal of achieving the best accuracy. The current benefits are it doesn't need a BIOS since it has easy configuration in regards to no custom settings to fix specific games, support for CHD disc images, straight forward design and a few others...

DobieStation is an experimental open-source PS2 emulator which aims to be easy-to-use, fast, and accurate on PC while also providing an optimized Android port. A large portion of the PS2's library can boot or get to menus and some can even go in-game, but it isn't recommended for general use yet. 

That said, it also requires a copy of the PS2 BIOS.

hpsx64 is a highly-experimental Playstation 1 & 2 emulator which can run quite a few commercial games, but probably at slower speeds than others. It's actively being developed and looks quite promising but isn't recommended for general usage yet.