What is a fan game and ROM hack?
A fan game is a completely original game (or a remake) that is created by fans of a franchise, series or particular official game. A ROM hack (or ROM patch) is modifying a game to alter the contents of that game's data and content. Technically a ROM hack can be considered a fan-game, but I'll exclude it for this.
I will not include mods for PC games, just to keep this focused.
ROM Hacks
Translation hacks
A translation hack is when you take a ROM and translate the game into another language. This has made many Japanese-only games playable in English. Some examples are:

Mother 3
The sequel to EarthBound, Mother 3 was originally a Japanese-only game released on the Game Boy Advance. It was originally meant to release to other countries but due to numerous legal and IP issues, it was not possible. This game is the most popular example of a translation hack, and has it's own website here. Without this translation hack, no one would know or care about the Mother series, and would just be remembered as that one quirky RPG for the SNES.
Nintendo has re-released this game on the respective virtual consoles, but it has been in Japan only. This translation hack is probably the most popular, and most people have heard of Mother 3 because of it.

Captain Rainbow
A one-off game by the Chibi-Robo developers, Captain Rainbow is a Wii game that released only in Japan. There is an in-progress translation hack here (these things take time!)
The game flopped hard financially but thanks to the ongoing translations, people can still experience this weird and "interesting" adventure.

Trials of Mana (Seiken Densetsu 3)
An indirect sequel to Secret of Mana, Trials of Mana was a Japanese-only game released on the Super Nintendo, and is considered one of the greatest games of all time. It was not localized for the longest time until fans made a translation hack here.
A rare case of games actually being officially localized, this got an official translation in 2019 in the Collection of Mana which fans ripped immediately, and even got a (pretty bad) remake also called "Trials of Mana" in 2020.
You can look at translation hacks on ROMhacking.net.
Content hacks
A content hack is ROM hacks, using the original game as a base to make whatever the creator wants.

Super Mario Star Road
A ROM Hack to Super Mario 64, and it is most probably the most recognizable Super Mario 64 ROM hack out there. It released in 2011 by Skelux and has all new levels, new music, new hub and new everything and 130 new stars to collect. You can view it here.
I won't include any more Super Mario 64 ROM hacks in detail because there is an endless treasure trove of them.

Relics of the Past
A ROM Hack to Breath of the Wild, it tries to make the game much harder, almost kaizo like. It boasts itself on not just buffing enemies like a lot of hacks, and actually makes modifications to the world (but it still uses the original world, a custom world would be an insane amount of work). You can look at the website here.
I wanted to show this example because there are multiple recent, modern games that also get ROM hacks. The scene is not just limited to old consoles.

Pokemon Unbound
A ROM Hack to Pokemon FireRed/LeafGreen. This is considered to be one of the best Pokemon ROM hacks (there are many out there) and some say that it is better than actual official games. Just because it is based on a GBA game, it does not mean it is crude, as it is filled to the brim with custom code and content. Forum post here.
Every single Pokemon from Generation 1 to Generation 7 are in this game. So you can imagine how big this game is to accommodate for everything.
Other games include:
Super Mario 64 (Nintendo 64)
- B3313 - Internal Plexus (2021) by ChrisRLillo and the B3313 devs
- Peach's Fury(2022) by Kaze
- Super Mario 74(2011) & Extreme Edition(2012) by Lugmillord
- Star Revenge series(2012-ongoing) by BroDute
- SM64: Last Impact(2016) by Kaze
- Super Mario 64 Land(2019) by Kaze
- SM64: Beyond the Cursed Mirror(2023) by rovertronic
- SM64 Sapphire(2017) by Nebula
- SM64 Decades Later(2024) by BroDute
Other Games
- [Super Mario Sunshine] Super Mario Sunburn(2020) by Wade - A revamp of Sunshine with new QoL features.
- [Castlevania Aria of Sorrow] Aria of Sorrow: Reprise(2024) by Xanthus - Transforms AoS into a rogue-like.
- [New Super Mario Bros. Wii] Newer Super Mario Bros. Wii(2013) by Newer Team - Basically takes the template and engine of NSMBW and makes unique good levels out of it, taking inspiration off of other 2d platformers.
- [Super Mario 64 DS] Another Super Mario 3D(2016) by Newer Team - The first SM64DS rom hack and the only major ROM hack out there for the game. Is pretty good.
Other Games (In development)
- [New Super Mario Bros. 2] Super Mario Bros. Next by Next Team - An all in one mario game using the engine of NSMB2. New levels, new world, new enemies, new everything.
- [Super Mario 64 DS] Super Mario 256 by mibts - Currently a demo but an extremely technically impressive hack in development. Considering 256 stars.
You can look at ROM hacks on ROMhacking.net, and ROMHACKING.com. (Yes, confusing, I know.) There's also things on GameBanana.
Fan games
Original games
In fan games, an original game is using the intellectual property of of a franchise to create a new concept, where it can have new characters, story and new gameplay.

Psycho Waluigi
A completely original fan game released in 2011 by Thunder Dragon, based around the idea that if Mario, Luigi and Wario all have unique games, Waluigi can have his own 2D platformer!
With elements from Yoshi's Island, Wario Land and other platformers, it tries to make Waluigi it's own fleshed out character. Instead of jumping or throwing enemies, you use psychic powers to control the enemies.

Pokemon Infinite Fusion
A Pokemon fan game based around the idea that you should be able to fuse any 2 Pokemon together. Every single Pokemon, together. This means that there are over 200,000 unique "fusions" in this game, and over 100,000 custom Pokemon sprites!
Granted, it isn't a new world. It's a modification of the Kanto area and a remixed Johto area. But still, it is a marvel that this even exists.
Other Games
- Paper Mario 3D Land(2014) by DJ Coco - A Super Mario 3D Land inspired Paper Mario platformer.
- Super Mario Flashback(2020) by Mors - A completely original very high quality 2D mario game, with elements from the 3D games. Is just a demo at the moment.
- Mario and The Music Box(2015) by Team Ari - A psychological horror visual novel game that stars Mario, inspired by stuff like Yume Nikki, Corpse Party etc.
- Super Mario 63(2009) by Runouw - The classic Adobe Flash game from back in the day. It tries to remake 64 in 2D and also takes stuff from other Mario games, creating an original game.
- Sonic Robo Blast 2(2009) by Sonic Team Jr. - A platformer game using a heavily modified version of the Doom engine, trying to recreate the 2D genesis games in a 3D environment. This is also less of a standalone game and more of a toolkit so other creators can use the SRB2 engine with, like Sonic Robo Blast 2 Kart, a kart racer.
- Castlevania: The Lecarde Chronicles 1(2013), and 2(2017) by Migami Games - The first game is a classicvania inspired game with elements from the Metroidvania titles, while the second is a metroidvania game. Has official Castlevania voice actors, amazing music and is considered one of the best in the Castlevania fan-made games and rom hacks.
- Pokemon Uranium(2016) by JV - One of the best Pokemon ROM hacks, and is more like a traditional Pokemon game.
- Mushroom Kingdom Fusion(2020) by Confuzyon - 2D platformer in the mushroom kingdom with different franchises all fused together. 17 different franchises.
Remakes
In fan games, a remake is taking an original official game and having a fan or a group of fan remake (or remaster) and re-imagine it for a new age.

Another Metroid 2 Remake
A remake of the original Metroid II on the game boy created by Milton Guasti / DoctorM64 and is considered one of the best Metroid fan games and is in the running of best 2D Metroid games with official Nintendo games. New content was added while staying true to the original game.
Straight after the release of AM2R, it was DMCA'ed by Nintendo, and then a year later Nintendo released an official remake of Metroid II called Metroid: Samus Returns. I would honestly recommend you play this over Samus Returns, and consider it for a second Metroid game (Play Fusion/Zero Mission first).

Sonic - Project 06
A remake of Sonic 06 by ChaosX, which tries to address the issue of the original game being considered one of the worst games of all time. Project 06 tries to fix all of the issues with the original game as well as restore scrapped content.
The game is not done yet and has only a few demos, but Sonic, Shadow and Silver's levels are all complete and playable.
Other Games
- Black Mesa(2020) by Crowbar Collective - A remake of Half-Life built from the ground up in the Source Engine. Valve actually allowed this on Steam, which is awesome.
- Rayman Redemption(2020) by Ryemanni - A remake and reimagination of the first Rayman game, which is a unique 2D platformer.
- Sonic Triple Trouble 16bit(2022) by Noah Copeland - A remake of the original Triple Trouble, with new 16-bit graphics and remade in a modern age.
Some screenshots from MobyGames.