Availability: 100% Free, Homepage

Everyone's favorite Gamecube/Wii emulator on Windows has an excellent iOS port. Enjoy games for these two consoles in full HD (1080p) with several enhancements: compatibility with controllers, turbo speed, networked multiplayer, and more! Check out its compatibility list to ensure the games you want to play fully work. It's a CPU heavy emulator, so it requires higher end devices to run well.

Availability: 100% Free

An N64 emulator for iOS and iPad. Powered by the Latch Controller, an innovative touch control system that lets you tap, hold, or lock inputs. Gather friends for seamless 4-player local multiplayer across four separate devices for true mobile “couch” co-op, with full support for 3DS games included right out of the box.

Availability: 100% Free

An SNES emulator optimized specifically for Apple devices, using Metal for video rendering and AudioUnit for low-latency audio performance. Key features include physical game controller support, customizable touch screen layouts, save states, cheat codes, turbo mode, and rewind functionality. Additionally, it supports external display output and utilizes iCloud for backing up and restoring user data.

Availability: Free with ads, Homepage

iGBA is a lightweight emulator designed to run Game Boy, Game Boy Color, and Game Boy Advance games. It features customizable touch controls, auto-save states, display filters, frame skipping, and beta support for Apple Watch. Additionally, it offers UI personalization through custom skins, color schemes, and AirPlay integration. The free version has ads. If you want to remove the ads, you need to pay the subscription for $0.99/month or $8.99/year (US dollars). If the ads annoy you and you refuse to pay, RetroArch works fine for GBA, too.
*iGBA is also available for iPads!

Availability: 100% Free, Homepage

A Game Boy emulator that supports Game Boy Classic and Game Boy Color. SameBoy has a lot of competition! With RetroArch, Provenance, and Delta supporting Game Boy, why should you care about SameBoy? Well now, where SameBoy shines is that it specializes in Game Boy emulation. SameBoy supports the Rumble Pack, Game Boy Camera, Super Game Boy, motion controls, and more. It supports a wide range of controllers such as MFi, PlayStation, Switch, and Xbox controllers. It supports iPad and even Apple Watch!

I have this here as a stub - iOS currently does not have any standalone emulators for Sega Genesis, Sega CD/Mega CD, Game Gear, or Sega Master System. However, these systems are fully supported on every popular multi-system emulator. They emulate these systems perfectly.
*Note for Sega CD emulation: Due to iOS security restrictions, emulators can't access the multiple files that make up a Sega CD/Mega CD game. You need to look for games in CHD format, which combines the game data and audio tracks into a single file. If your Sega CD game is in ISO/BIN format with 10+ separate track files, it's not going to work.

Availability: Free with ads or $4.99 US, Homepage

Yaba Sanshiro's Saturn emulator (also known as uoYabause) is excellent. Game compatibility is really good. I recommend reviewing the compatibility list before trying out games. It's a CPU heavy emulator; it may not run well on slower/older devices. Don't forget to download and configure the BIOS (which is right below). This light version is free with ads and only allows installation of three games. The full version costs $4.99 US. If you hate the light version and refuse to pay for the full version, RetroArch is a good (free) option for Saturn emulation.

Availability: Free with ads

A simple, good PlayStation 1 emulator that offers save states, custom controller skins, and cover artwork for titles. The only annoying thing about it is that you're presented with fullscreen ads as you use it. If that annoys you, can you use RetroArch for PlayStation 1 emulation.

Availability: 100% Free, Homepage

The iOS port of the #1 open source Sony PSP emulator! It's a superb emulator with many features. PSP is rather CPU-heavy to emulate, so don't expect games to run perfectly on your iPhone. Fortunately, PPSSPP offers lots of options to help you optimize performance to help games run smoother.

Availability: 100% Free

A solid arcade emulator for iOS. It supports iPhone and iPad. MAME version 0.278 is the basis for its emulation. You can hook up popular controllers like Xbox, PlayStation, or Switch, or just use the touchscreen with features like a built-in lightgun option. There are even extras like save states, cheats, and visual enhancements to give classic games a fresh look.

Availability: 100% Free, Homepage

This is a mass system emulator with support for pretty much everything: PlayStation 1, N64, Nintendo DS, SNES, NES, Game Boy, Game Boy Advance, NeoGeo Pocket Color, Virtual Boy, Genesis/Mega Drive, Sega CD, 32X, Sega Master System, Game Gear, PC Engine/TurboGrafx16 (plus CD support), Wonderswan Color, Atari Lynx, arcade (MAME), and more. It offers lots of features: screen filters (shaders), save states, different screen overlays, cheats, netplay, and more! It's highly customizable! I highly recommend RetroArch.
*RetroArch is also available for iPads and Apple TV!
*Need help? I don't have a specific tutorial for RetroArch on iOS, but you can check out my RetroArch for Android tutorial. Both versions are nearly identical.

Availability: 100% Free, Homepage

Provenance is a multi-system emulator that supports the same amount of video game systems as RetroArch (which is pretty much everything). As you'd imagine, that makes them close competitors. Where Provenance shines over RetroArch is that it has a more visual UI. The UI also more closely follows that of iOS. Its Nintendo 3DS emulation is good as well. However, Provenance doesn't let you pick from a variety of different cores like RetroArch does.
*Provenance is also available for iPads!

Availability: 100% Free, Homepage

This is a mass system emulator with support for NES, SNES, N64, Game Boy/Color, Game Boy Advance, Nintendo DS, and Sega Genesis. Features include save states, cheats, gamepad support, and beautiful overlay skins for the onscreen buttons. You can sync your games, saves, and skins with Google Drive or Dropbox.
*Delta is also available for iPads!

Availability: 100% Free

I don't need to post a list of emulators for Mac OSX because you only need one: OpenEmu! It's a multi-system emulator that supports around 30 systems! It has a nice and clean UI, too, as expected from the reputation of Mac apps. Make sure to check out its wiki for help and to explore all the features available.