1st Place - epsxe
2nd Place - pSX
3rd Place - PCSX
2nd Place - pSX
3rd Place - PCSX
Well, at least according to Emulator-Zone. I've personally tested all 3 and the emulator that has impressed me the most is actually the number 2 spot pSX.
What is the difference you ask? If your an experienced emulator user, you might not have the need to even read this post! Just in case some of you want to play the classic PSX games and have recently looked into emulators then let me guide you.
ePSXe is known as the best for this reason, compatibility and customization. There are tons of plugins you can use, not just to play the games smoothly, but to enhance the graphics as well (Imagine playing FFVII in HD). Plugins that make use of the full potential of your PC's CPU and GPU. You've gotta admit, the PSX had very weak graphics compared to the new generation consoles nowadays. So, if your PC is capable, why not enhance the graphics because you can? Off course, there are still many other plugins other than just graphic enhancement, but for end users who just want to play the games. Those details are best kept at bay.
pSX on the other hand is 100% about compatibility! There is no plugin I know that enhances the graphics for the games in this emulator. It is light and emulates the console almost 100% if not already 100%! If you want to play the games the way they are suppose to look like on Television, then this is for you! There are no plugins to configure or at least easy enough to configure (granted you can read and understand terms like "Full-Screen, resolution...etc"). A good feature about the emulator is that it doesn't need a high-end PC to function well. I was able to emulate games with minor lags with a crappy netbook!
for PCSX, I haven't fully used it as much but I think it may have minor compatibility issues or just a unattractive user-interface to fall to 3rd place (it seems to function fairly similar to epsxe), but it's successor the PCSX2 is the best PS2 emulator around. Now that you know the basics of these emulators, feel free to try the out yourselves!