Achievements support

If you want to play older games with achievements there are a couple of ways. Either by playing with a modified emulator and fan-created achievements, or games that have been re-released on Steam.

See each Wiki Category Consoles, Computers and Arcade for individual dedicated system pages to see provided up-to-date listings and in-depth information on specific aspects like hardware features, peripheral support, compatibility and also enhancements features such as Achievements support etc.

RetroAchievements

This is the download page for customized emulators with achievement support included, as well as frontends and standalone emulators. Achievements will appear as overlays during gameplay. The source code is available here. You can actually develop sets by yourself. The full list of games is here.

Emulators supporting achievements:

BizHawk: Multi-system emulator

Dolphin: GameCube (NOTE: Requires a development version of 2407-68 or higher)

DuckStation: PlayStation

PCSX2: PlayStation 2

PPSSPP: PlayStation Portable

RALibRetro: Multi-system emulator based on libretro

RAMeka: ColecoVision, Master System, Game Gear and SG-1000 emulator based on MEKA

RANes: NES emulator based on FCEUX

RAP64: N64 emulator based on Project64

RAppleWin: Apple II emulator based on AppleWin

RAQUASI88: PC-8000/8800 emulator based on QUASI88

RASnes9x: SNES emulator based on Snes9x

RAVBA: Game Boy, Game Boy Color and Game Boy Advance emulator based on VisualBoyAdvance

RetroArch: Multi-system emulator

WinArcadia: Emerson Arcadia 2001, Elektor TV Games Computer and Interton VC 4000 emulator

Misc:

Exophase: A social network for people who play games from different platforms and software. Exophase specialize in trophy and achievement tracking, complete with dedicated profiles, gamercards, multi-platform leaderboards and more. Creating a profile to easily keep tabs on your gaming activity and share updates with others through social platforms like Twitter. In addition, meet other gamers through our growing community forums, tackle challenges together, and share tips. Exophase.com integrates with RetroAchievements, allowing you to track your achievements for retro games on the platform as well.

Steam Re-releases

Some older games get re-released on Steam with achievement support, like the Final Fantasy games.

One recent re-release of N20 for the PS1 on Steam was found out to be an emulated PS1 ISO bundled with a modified build of PCSX-Reloaded (using HLE BIOS) to support Steam Achievements. Tweaked, it could be the basis for a new build of PCSX-R with achievement support. The source can be found here.