For PS1 you have several good options. In descending order of flexibility (but arguably ascending order of ease of use):
PCSX-R:
Plugin-based open source emu. Good compatibility, decent accuracy. Requires the right plugins and setup to get running, but does allow stuff like resolution upscaling and some other graphical options/hacks, making it quite flexible. Has several branches that offer additional functionality like PGXP for even more flexibility.
ePSXe:
A plugin-based emu like PCSX-R, but closed source. One of the oldest PS1 emus still in development. Similar compatability and accuracy. Offers mostly the same functionality, but its closed source nature means make it slighly less flexible and a little behind compared to PCSX-R. Still a reliable choice tho.
Mednafen:
Lacks a GUI (tho front-ends like Mednaffe are available), but is very accurate and compatible, maybe even more so than Sony's own PS1 emu solution. Probably the closest you can get to playing on real hardware. Also its downside, since it offers no options for upscaling or tweaking.
Beetle PSX:
Basically the same as Mednafen, but as an emulator core for Retroarch, a multi system front-end that allows you to easily install, manage and run a whole bunch of emus from a single executable.
In short, Retroarch (Beetle PSX) if you want to get started quickly and easily. Mednafen if you want the most accurate experience. PCSX-R or ePSXe if you want the flexibility to tinker.
As for PS2, as already mentioned in this thread, there is only one real option:
PCSX2:
Plugin-based open source emu. Accuracy is only moderate, so expect minor audiovisual glitches. Still, it runs pretty much every PS2 game from start to finish. Plugins require some setup and fiddling, but when you know what you're doing, you can run games at high resolutions with anti-aliasing and anisotropic filtering. Basically a free HD remaster.
Also, PCSX2 actually has PS1 backwards compatibility, like a real PS2. Still better to use a proper PS1 emu tho.
