Matrix Raining Code

You can tell it's evil because it's Borg Green.

That trippy raining code in The Matrix that is constantly changing has pretty much become synonymous with high tech computing, VR environments, AIs, and Robots. Originally, it represented the ever changing nature of the Matrix and the overwhelmingly complex incalculability of it all. So when Neo starts seeing things in Matrix-Text-O-Vision, it represents him seeing and being able to manipulate the underpinnings of reality. Nowadays, it's mostly used to convey "Look, high tech computer with arcane, indecipherably alien power and mentality!" or as a Stock Parody for The Matrix.

"Traditionally", the Raining Code will be green, flow from top to bottom, consist of normal letters and numbers with what looks like Japanese kana or other strange runes, and change text mid fall. Newer adaptations will likely use a different color, direction for it to scroll, and text palette. Note that the original Matrix version used katakana characters and arabic numerals, mirror-flipped to obscure their shapes. The characters will generally leave "ghosts" on the screen as they fall, reminiscent of old Apple II RGB monitors, symbolizing the Ghost in the Machine, of course.

Incorporates Mojibake as a matter of course. Often displayed with Stat-O-Vision and in the Holographic Terminal. Compare/See also Design Student's Orgasm.

Examples of Matrix Raining Code include:

Advertising

A home security commercial showed a house being enveloped in blue and white 01's.
- If it's ADP, the original ad came before The Matrix.
The "traditional" Matrix-style code most recently showed up colored white in a makeup(?!) commercial.
White code with red highlights are used in the more recent Droid cellphone commercials.

Anime & Manga

The .hack series has something similar, with floating symbols and screen static in the corrupted areas.
The Eureka Seven series has Renton's Compac Drive, with the name Eureka and other minamul streaming et cetera.
For obvious reasons it occurs every so often in the Digimon franchise; it was quite common throughout Digimon Tamers, and in Digimon Savers it plays a part in the transformation sequences.

Films -- Animation

The Matrix's raining code was ripped/homaged from the Ghost in the Shell opening scene, which didn't rain, but did change characters, and flowed across the screen. Note that these characters were a machine-code translation of the English-language credits that they transformed into.

Films -- Live-Action

The Matrix, of course.
Star Trek: Nemesis. The Federation database is depicted as blue matrix code
In another homage to Ghost in the Shell, Avalon (film)'s opening credits are orange falling text-like.

Literature

The Bible Code books have this on the backgrounds of the covers.

Live-Action TV

Andromeda has this in their VR scapes.
Battlestar Galactica: Notable in that the Cylons have it on their computer screens, data streams, and even as holograms projected onto the set. An interesting tidbit is that the first hybrid who is not a Mad Oracle (but oracular and sane) has blue rather than the typical cylon red for his code holograms.
Caprica carries on the Cylon example, and explains it as being a human (well, colonial human) programming language, specifically the one Zoe used to create her AI. Here, it's orange/red, rains upwards, and when it hits the top row it gets pinballed to the right where it's presumably "executed".
An episode of Numb3rs has the computer in a DARPA sponsored project to develop an AI with Matrix code showing on its displays. This might have been a partial subversion, since a lot of the what the computer was doing was eye-candy to impress the folks footing the bill for it, and the actual "AI program" was just an overblown version of "Eliza".
The Wraith of Stargate Atlantis use this on their organic computers.
Animal Planet's The Most Extreme used this for the background (along with a healthy dose of "digital green"). One can only guess that Rule of Cool was in play.
Kamen Rider Double notably uses this to portray the interior of the Xtreme Memory, which rescued Philip from certain death at the hands of the Weather Dopant. Bits and pieces of code tend to appear during the Xtreme transformation too, but not nearly as much as Xtreme Memory's inside.
In the fourth season CSI New York episode, "The Thing About Heros", the team is trying to analyze data on a broken mp3 player. The image displayed looks like raining Matrix code.
No Ordinary Family used this to represent JJ's super brain powers as used to crack a file encryption (the code was yellow instead, as with most representations of said power in the show.)

Music

"Weird Al" Yankovic does this with the video for the song "Virus Alert". It seems to consist of his name, 1234567890, and Leet Speak.
"Digital Rain" from the band Star One. The song is an homage to The Matrix

Video Games

Enter the Matrix, The Matrix Online, and The Matrix: Path of Neo, naturally.
It was a requirement that at least one stage of any Mega Man Battle Network game had to have white, future-y roads and a background with green 0 and 1. It sometimes scrolled, sometimes blinked.
The background of the MASON System in Apollo Justice: Ace Attorney.
Deus Ex has a "Matrix mode" cheat that turns the entire game into scrolling green text.
- Timesplitters: Future Perfect had a similar cheat code available for the multiplayer deathmatch mode.
- The second Bejeweled game also has such a cheat; typing 'network' during a game would convert the background to Matrix Raining Code.
Zone of the Enders: The Second Runner has the tutorial area VR environment show whisps of code occasionally.
The loading area for Assassin's Creed has chemical formulas as well as other random bits of text floating about.
- And coding conveniently "leaks" into the main game to highlight important stuff.
- Subverted in Assassin's Creed Brotherhood, where the flickering snippets of binary mean it is actually literally raining real code. Sorta.
The screensaver in Zelda Classic is this.
Used in a few places in World of Warcraft, of all things. Most notably, surrounding one of the bosses in the Nexus.
When an item is placed on a laboratory analyzer in Startopia, scrolling green text appears around the object on the table. Zooming in on the analyzer shows that the text is actually the name of the object being analyzed.
While you spent most of the area indoors, if you looked closely while standing outside in Sanctuary in Metroid Prime 2: Echoes, you would notice that it's raining. Look even closer, and you'll see its Matrix Raining Code, raining skyward.
Taken to its logical extreme in ASCII Quake.
Devil Survivor's genius programmer Naoya wears a Badass Long Robe with Matrix Raining Code printed on it.
Final Fantasy IX features the code around teleportation gates. It works as foreshadowing for the more technologically advanced second half of the game, but it's still slightly jarring.
These were to appear in the second Knights of the Old Republic game, but the planet they were on, the droid world of M4-78, was cut from the final product. A handful of mods exist that allow the player to visit what maps exist of M4-78, though.
The intro cutscene for Bayonetta has some kind of red-Matrix-Raining-Code thing going on when the title of the game is being displayed. Why? Rule of Cool.
Cortana from Halo is represented as a hologram made of purple Matrix Raining Code and Tron Lines.
Appears at the beggining and end of the 4th cutscene of Crash: Mind Over Mutant due to the fact that Crash and Coco have just put their NVs on.
The Scrin campaign's cutscenes in Tiberium Wars usually feature alien glyphs raining somewhere on the screen, explained as being the Unusual User Interface of the Scrin mothership. A notable example in the first cutscene is a horizontal string of glyphs shuffling and changing into Latin letters as the ship's AI is analyzing Earth's radio transmissions (the very same ones shown in various cutscenes in the other campaigns!) until it reads "LANGUAGE ASSIMILATION COMPLETE" in plain English; from that point on, all relevant messages appear in English.

Webcomics

As part of a larger Shout-Out to The Matrix, Sinfest represents omnipresent sexism as raining green letters, spelling out words like "slut" and "whore."

Western Animation

Seen in Futurama, "Love and Rockets", when Bender merges his programming with the Planet Express ship.
Code Lyoko has constant raining blue "0" and "1" inside squares in the Lyoko Towers, the tunnels between the Sectors or over the surface of the Celestial Dome in Sector 5. Those squares can also be seen flowing on many background computer screens in the real world, colored green and looking more like classical Matrix Raining Code.
The 3rd Season DVD of The Simpsons has this as one of the Couch Gag menu transitions.
For a while in Transformers: Beast Machines, Megatron manifested as a big glowy head made out of this type of code, albeit in Cybertronix
Justice League Unlimited episode "The Once and Future Thing, Part 2: Time Warped" had Bruce Wayne of the Batman Beyond future trying to calculate the damage to the timeline caused by the villain Chronus' very irresponsible time manipulation. The digital readout was essentially the matrix raining code.
In the episode "Don" from Regular Show, the government uses this to engulf the entire park when Rigby messes up the audit.
Chaotic has a lot of this. When traveling to the game world, characters pass through a tunnel made out of swirling letters and numbers, and when a creature loses a battle, they dissolve into exploding letters and numbers. Justified, since the code is actually a plot point (sorta).

Real Life

One of the old DOS viruses (Cascade) caused random letters on the screen to "fall".
Matrix Raining Code also resembles a scrolling hex dump that typically represents binary code in a somewhat human-readable form. It won't be too much of a stretch to assume that large amount of code may require people to use some more condensed representation.
Another possible origin is a tendency of programmers to prefer text-based interfaces (such as Unix shell) and to use text printout for debugging (often in cryptic format, to conserve resources and minimize impact on the running program).
Used at the start of the descent of the 2007 version of Epcot's Spaceship Earth, with bopomofo instead of katakana.
And of course present in tons of Matrix-style screensavers of all variations for most operation systems, from Win XP to Linux/UNIX.