EmuNations

DuckStation is an open-source PlayStation 1 emulator mainly for PC and Android. It's still in early development but aims to have system simulation, debugging and resolution upscaling. It can play multiple commercial games albeit not at full-speed nor without any glitches. It has Qt/SDL front-ends an, D3D11/OpenGL hardware and software rendering and many more features. Overall, its outlook seems quite promising!

Check here for Duckstation builds with netplay!

Note: A real bios image is required. 

My Emulator Doesn't Need A Frickin' Excellent Name is an open-source, multi-system emulator, driven from the command-line, OpenGL and SDL. Its Sony PlayStation, Sega Saturn, Nintendo Virtual Boy, and NEC PC Engine (TurboGrafx-16) original cores are recognized for their high quality, compatibility and accuracy.

There are several front-ends available, if you don't know how to execute via command-line:

• MedGui Reborn (Windows)

• MedLaunch (Windows)

• Mednaffe (Multi-platform)

For CRT screens, you can try emu4crt since this mod is intended for ~15kHz TVs or arcade monitors. Check it out here.

PCSX-Reloaded uses the PSEMU plugin interface to provide most functionality. It's currently integrated with PCSX-R via Pete's OpenGL v1.78 plugin and Tapeq's Tweak extension to Pete's OpenGL v2.9 plugin. PGXP (Parallel/Precision Geometry Transform Pipeline) is an enhancement for PlayStation emulation that produces high precision fully 3D geometry data that was not available on the original console hardware.

Features of this build:

High Precision Vertex Data (more stable geometry)
Reduced Triangle Culling (more detailed models)
Perspective Correct Texture Mapping (reduced texture distortion)

Note: It has a very capable Internal HLE BIOS that can run many games
without problems. That said, a real bios can improve compatibility, especially with certain games and with the use of memory cards.

It's a modern experimental PlayStation 1 emulator which focuses on relatively modern hardware (2010+) and non-legacy OSes. Avocado is currently single-threaded, so single thread performance is what really matters since multi-core won't bring much benefit. Also, it currently requires OpenGL 3.2 but will be lifted once software rendering is supported.

Note: It requires a real console BIOS and has to be stored in the data/bios directory.

A previous contributor has forked the Playstation Emulator PCSX with more recent addons such as PGXP and lightrec recompiler. Goal is to clean it up and right the wrongs of the past while trying to dissociate itself from plugins.

XEBRA is a closed-source Japanese PlayStation emulator which focuses on accuracy. That said, it does not use any plugins and only outputs in native resolution.

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Note: It has a very capable Internal HLE BIOS that can run many games without problems. That said, a real bios can improve compatibility, especially with certain games.

eNGE is a pure JavaScript Playstation 1 emulator which runs reasonably fast and through Google Chrome. It has basic functions such as BIOS support, gamepad, loading cdrom, memcards and even settings for increasing rendering resolution. That said, it doesn't have other advanced rendering and customization features compared to others. 

ePSXe (enhanced PSX emulator) is a closed-sourced emulator based on PSEmu Pro plugin system which emulates the GPU, SPU and CD-ROM drive. It's able to run most PlayStation games quite accurately without having to alter extensively through trial by error testing.

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Note: It has a very capable Internal HLE BIOS that can run many games
without problems. That said, a real bios can improve compatibility, especially with certain games and with the use of memory cards.

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.