Episode Zero: The Beginning

Well math says that 0 comes before 1, so putting "Zero" in the title of a work, or episode of that work, can be a good shorthand that a work is a Prequel, prologue, or even a remake of a first chapter. Some comic book origin stories are shown in issues numbered 0.

Of course if "Zero" is in the title for any other reason, it doesn't count (as it wouldn't be the beginning). So works like Mega Man Zero and Phantasy Star Zero wouldn't be an example.

Also, the names have to be official, not fan nicknames.

Compare Numbered Sequels, Unusual Chapter Numbers, Year Zero.

Examples of Episode Zero: The Beginning include:

Anime and Manga

The Haruhi Suzumiya episode "The Adventures of Mikuru Asahina Episode 00" (though that's more of a subversion, since the episode mainly comprises the SOS Brigade's school movie that's produced in the middle of the series, and will be completely incomprehensible to anyone who isn't familiar with the show - and not much more comprehensible to those who are).
Macross Zero
Ga-Rei Zero
SD Gundam Force Emaki Musharetsuden Zero was a prequel to SD Gundam Force Emaki Musharetsuden.
Rose Hip Zero is a prequel to Rose Hip Rose.
Mai Otome 0 Sifr, a prequel to the other Mai-Otome films.
Steel Angel Kurumi 0, a prequel to the Steel Angel Kurumi series.
Aika Zero, a prequel to Agent Aika.
Mobile Suit Gundam Wing: Episode Zero.
The film Un-Go: Episode 0, prequel to Un-Go.

Comic Books

Robert Crumb's Zap Comix came out with an issue zero: he had lost the art to the first issue and made a new one, then discovered the missing artwork and released it later.
Spider Girl's first issue was numbered 0, and was a reprint of the What If issue she first appeared in.
"Zero Month" following DC Comics' Zero Hour is a borderline case; the #0s all retell the heroes' origins, but in Flash Back, with a Framing Story in the present.
Marvel Comics had a "Flashback Month" in which every comic published an #-1, set before the origin stories.

Film

Cube Zero is the third film in the series but is a prequel to the first film.
Yoshie Zero, a prequel to Mutant Girls Squad. A rare example where the non "zero" part of the title doesn't match the sequel.
Ring 0: Birthday. A prequel to the Japanese film Ring.
Ichi the Killer: Episode Zero, an animated prequel to the live action film Ichi the Killer.

Literature

Fate/Zero

Live Action TV

Pretty Guardian Sailor Moon Act Zero, a prequel to Pretty Guardian Sailor Moon.

Video Games

Resident Evil Zero
The Street Fighter Alpha series, which eventually became a remake of Street Fighter, is called Street Fighter Zero in Japan.
Metroid Zero Mission is a remake of the first Metroid game.
Zork Zero: The Revenge of Megaboz. Although it was the ninth and last Zork game released, it's actually a prequel to the first eight Zork games.
Perfect Dark Zero, a prequel to Perfect Dark.
Ace Combat Zero: The Belkan War, prequel to Ace Combat 5 The Unsung War.
Record of Agarest War Zero, prequel to Record of Agarest War.
The Dating Sim Dream Club Zero, a prequel to Dream Club.
Silent Hill Origins, which is called Silent Hill Zero in Japan.
Final Ninja Zero is a prequel to Final Ninja.
Dark Void Zero. Originally starting out as an April Fools joke based off of an 8-bit rendition of Dark Void's main theme, the game soon became a fully-fledged DSiWare title claimed to be a "legendary 'lost project'" of Capcom's, recently found locked away in their vaults.
Samurai Shodown V, a prequel to the original game, was titled Samurai Spirits Zero in Japan.
While "Ultima 0" was originally just a Fan Nickname for Akalabeth, the title was made official when it was included in the Ultima Collection compilation box.
Mother combines this with Sequel First. It was originally localized under the title "Earth Bound", but was never officially released outside of Japan. Its sequel was later released under the original's intended title, and so leaked copies of the original were modified so that the title screen said "Earth Bound Zero".

Web Animation

24: Day Zero. A series of animated webisodes. It's a prequel series, focusing on Jack Bauer's first 18 months at CTU.

Web Comics

On the Origin of PCs, the first The Order of the Stick prequel, was published as issue number 0. Start of Darkness, the second prequel, takes place even earlier, so it's issue number -1.