First off, i've heard doubts of emulators for Sony Emulators Existing, I want everyone to Know that Sony PlayStation 1, 2, 3, and PSP have all been emulated to some extent. Now, on to other things.
PS1 Emulation: I recommend getting ePSXe, AND PCSX-Reloaded, though ePSXe is said to be better, PCSX-Reloaded can play certain games ePSXe can't, get both emulators and the games you want, extract the contents of the games to get a file that the emulator can play, configure the controls of ePSXe, and PCSX-Reloaded, change the control plugin in PCSX-Reloaded to Lilypad, I'll tell you how to get lilypad later, now test the games to see which ones work on which emulator the best. After testing, play the game you want and enjoy!
PS2 Emulation: get PCSX2 as it is BY FAR the best PS2 Emulator, get the games you want and extract the files of the games to get an ISO file, configure the controls and enjoy the games, oh and one more thing, PCSX2 comes with the lilypad plugin, send the plugin to PCSX-Reloaded so you can use analog and vibration.
PSP Emulation: There are several PSP Emulators, some can run games others can't, but I recommend PPSSPP, once you get it, configure the controls and get the games you want, then extract the games to get and ISO file, run the file in PPSSPP and enjoy!
PS3 Emulation: there is currently only one PS3 Emulator in Existence, Rpcs3, it can run about 5 games, none of them are good in my opinion, the games run EXTREMELY slowly on even the best PC's, and the ISO files are hard to get, while you can download Rpcs3 and the ISOs, you'll need a Computer with at least 5 cores, each running at 9Ghz, experiment if you want and maybe you'll get the games to run at a playable speed, and maybe you'll enjoy them, just if you get ISO files, tell me, I've been looking for them.
PS4 Emulation: No such thing, if you see a site that claims to have a PlayStation 4 Emulator, don't fall for it, it's bullshit.
PSVita: Same as PS4
That's about it, hope this helps!