Coming from the Latin ludo "play" and narrative-- narrative, Ludonarrative Dissonance is a recent(ish) critical term to arise in the advent of game criticism becoming more of a thing. As video games become more recognized as an art form and as a valid way of telling a story, doors have opened to more critical looks at games in aspects of mechanics/gameplay, art, and story, and how these aspects of the game come together as a cohesive whole or. . . don't.
Ludonarrative dissonance specifically refers to when the gameplay mechanics of a game seem to contradict or are otherwise disingenuous with the story of the game. The term was coined by Chris Hocking, former creative director of LucasArts, in regards to the game Bioshock and the mechanics of choosing to harvest or save Little Sisters versus the main story wherein you must help the city Atlas. According Hocking, the gameplay mechanics of Little Sisters conditions the player that self interest is the way to progress and gain power, but that the story tells the player the conflicting message that helping others is the way to progress. In the mechanics of the game, you can choose to help others or to choose self interest. In the story, you can choose to help others. . . or turn the game off.
This problem isn't an uncommon one in games. Everyone and their grandmother has made the joke that Pokemon, a franchise built around supernatural, government-approved dog fighting, continuously espouses messages of peace and love between human and Pokemon-- to the point where the franchise itself bends over backwards with every new iteration to try and defend or soften the implications. Likewise, any game where the story dictates to you that you are the chosen one hero and paragon of goodness, but then lets you mass murder an entire city's worth of people without the story acknowledging it is suffering from dissonance.
This kind of dissonance also comes in smaller scales. There's a funny example in Watch Dogs where a character is visiting the grave of a loved one that sets up the game's plot, and then once the cutscene ends, the player is given the ability to climb on top of their loved one's grave. In Skyrim, the world is going to end because of the resurrected dragons unless you do something about it, but you can run off into the wilderness and steal cheese from farmers for 50 hours instead. The main narrative says that time is of the essence, but the open world gameplay says it's not important.
In contrast, the ludonarrative in Shadow of the Colossus is hailed as being near perfect. The protagonist, Wander, must climb the aforementioned colossi and the gameplay makes the player feel small and quick as Wander struggles to keep his grip. Some of the colossi are big enough to become terrains, and the gameplay in the battles makes it feel like it. When the player is turned into a giant behemoth, the gameplay makes the controls sluggish and heavy, emulating the slowness and strength of the colossi Wander had fought before. And when Wander regains control of himself in a hopeless escape attempt, the game gives control back to the player and lets them struggle before sucking them into the void. The gameplay and story match up perfectly.
The problem with ludonarrative dissonance as a term for criticism is that people can't seem to agree how much of this kind of criticism is a valid way to criticize games, and how much is just nit-picking at inherent qualities that video games have just by their nature. The other problem is that a lot of people get confused between "game play is dissonant with the narrative" and "bad storytelling".
If you can think of any examples, shoot me a message and if they fit, I'll include them.
Glowing Fish: Ludonarrative Dissonance: Back in the days of the "List of RPG cliches", one of the examples was the "Hard/Soft Rule" of death. Games where resurrecting allies multiple times in a combat round is a game mechanic will have characters stay dead when they die for plot reasons. "Why not give Aerith Gainsborough a Fenix Down". Also related: characters can have a dragon breathe flame in their face and lose half their hitpoints, but in a cut scene, a single knife thrust will kill them permanently.
Estelore: I know you already mention pokemon, but the latest versions, Ultra Sun and Ultra Moon, have an hilarious dissonance regarding timing and perceived urgency. There are several areas that use tense, urgent ambient music during specific points in the game (while having 'elevator music' that deliberately soothes during other times in the same locations), but these locations are sprawlingly huge and full of hidden features. As a result, the character winds up running around enormous complexes, searching every trash basket for hidden items, deliberately confronting battle opponents for prize money that isn't available during peaceful times... when the mission is supposed to involve stealth and quick movement toward the objective. And all throughout, that ominous, clock-bending music keeps playing. The game even makes it possible to completely leave the location in the middle of the mission and return to it.