When Dimensions Collide

This trope is about when two worlds exist and the physical laws or geometries of one bleeds over into the other.

This usually takes one of two forms:

A world with overly simple rules and internal logic which is entered by somebody from and Earth-like world. These works are more prone to humour as the native inhabitants fail to understand or loudly disbelieve things which would be obvious to normal humans.
A world which has an alien set of physical laws (or somebody from it), that then interacts with an earth-like world. Type II worlds are almost always in the horror genre. Visitors from these worlds are often either The Fair Folk or Eldritch Abominations.

Importantly although this sometimes overlaps with alien geometries it is not about the alien shapes involved, (although they can be a side-effect) but rather the idea that something is invading a world sending their laws of physics first.

The litmus test for being this trope is:

That there must be two independent worlds.
Something must cross from one world to the other.
The "laws" of the invading world must work in the invaded world. (for example if a mage comes from magical world A to mundane world B it only becomes this trope if the mage can the cast magic in the mundane world were magic is normally impossible)

Examples of When Dimensions Collide include:

Type I

Simple worlds invaded by "real world" elements

Film

In Pleasantville, the protagonists are high school students from our world who enter the world of a 1950s TV serial entirely in black and white, in which everything is excessively pleasant (as well as lacking colour there is no rain, crime, homelessness, fire, sex or toilets). Throughout the film their actions impact the world around them and colours and concepts from the real world (like fire, sex, colour and rain) start to appear as a result.
Cool World has elements of both types; the Type I example is the humans transported to Cool World.

Literature

In Flatland, A Square is visited by a sphere from the mysterious dimension of "up" and interacts with the strange world of the third dimension. A sphere manifests as a circle that grows and shrinks, able to bypass all flatlander doors and walls and even touch "inside" a flatlander. The protagonist eventually learns to think multi-dimensionally and is considered insane by most.

Webcomics

Erfworld: The world's rules reflect those of a traditional turn-based tabletop game. Particuarly notable because the protagonist experiments with the physical laws of the world in an effort to better understand the rules of the game and how to cheat with them. However the protagonist himself appears to have several interesting and unique properties that are a hold-over from reality (such as a lack of visible stats, normally an impossibility).

Type II

Earth-like worlds invaded by alien elements

Film

Jumanji: The flora and fauna of Jumanji that invades the "real world" is able to do things that would be impossible for real life equivalents (monkeys that can ride a motorcycle, plants that can grow incredibly quickly, a Pelican that can fly the board game without trouble etc...)
Last Action Hero is a quasi-example. On the one hand, when an action hero crosses over from the world of movies into the real world, he nearly dies from a gun shot he'd normally shrug off as Just a Flesh Wound; similarly, a villain who makes the same leap is delighted to discover that in the real world the bad guy winning is actually possible. However, when Death crosses over into reality from the movie world, he's still able to kill people just by touching them. We also see that, when the movie hero shoots the movie villain in his cybernetic eye, it causes a ridiculously huge explosion, even when they're both in the real world.
As noted above, Cool World has elements of both types; the Type II example is Holli Would's attempt to become a human, and the affects it has on the "real" world.

Literature

The stories of HP Lovecraft often involve aliens that dwell in more than the traditional three dimensions and who occasionally interact with Earth. The twisting of logic and geometries involved by interacting with these alien space gods usually drives people mad as they are unable to comprehend them.
Isaac Asimov's The Gods Themselves is a story based around a new form of energy that arises when people discover how to exchange matter between parallel universes. The protagonist slowly realises that as they exchange matter between worlds some of the cosmological constants also change very slightly, but with potentially apocalyptic consequences.

Live Action TV

In Fringe, the space-time continuum is breaking down due to contact between the two universes, which are implied to have slightly different physical laws (the show is inconsistent on that point, though). Interesting as the bleed appears to be two-way.
The climax of season 5 of Buffy the Vampire Slayer involves the literal breakdown of all dimensional barriers as Glory tries to make her way home; all manner of demon worlds begin to bleed over and into Sunnydale before the rift is closed.

Manga

In Uzumaki, the inhabitants of a small fairly isolated town begin to notice a repeating spiral pattern that manifests in a number of disturbing ways. People, objects, plants, Galaxies, Space and time eventually twist into a spiral shape drawing the inhabitants in. The closing scenes show the spiral world below and the narration suggests that this spiral world invades the mundane on a regular basis, leaving only ruins behind when things return to normal.

Tabletop Games

The Witchcraft supplement "Armageddon" is described as having elements of this in areas that are captured by the enemy, such areas are changed radically into something alien and inhospitable to normal life as we know it. Victims end up fused together in collective masses of flesh and otherwise twisted beyond all recognition.
The game Torg is explicitely this trope, in which Earth is invaded by a number of other worlds that each have their own genre-like set of laws (which are spelled out in detail for each area, so in the pre-historic themed North America technology does not work and groups of people devolve into small tribes, whilst in the Pulp themed Middle-East people people drift into stereotypes, their allegiances become easily changeable and good triumphs over evil). The playable characters are those rare individuals who are able to carry their own native laws of physics around with them and exercise them against others.
Shadowrun has an element of this when magic (re-)enters the world, changing the limitations of what is possible.
Dark Conspiracy. Parts of the U.S. have been taken over by Dark Minions invading through portals from their home dimensions. These areas are known as "Demonground", and they're filled the corruption flowing from the portals. Common elements include bizarre vegetation and weird organic tunnels.
Witch Hunter: The Invisible World. Hellpoints are direct doorways to Hell itself. The areas around them are filled with malign influence and evil creatures. Demons may easily enter the world at these places.

Video Games

In The Longest Journey, the technological world of Stark and the magical Arcadia are usually well-insulated from each other. However, when the Balance between them begins to falter in the beginning of the game, weird stuff begins to happen in both worlds, such as a TV show about rainforests transporting the viewers into an actual rainforest, or a handheld calculator trapping a mage tampering with it inside.