Batsignal

Origins
The fictional origin of the signal varies between timeline and media. It made its first appearance in Detective Comics #60, February 1942. In the 1989 Batman film, Batman gave the signal to the police as a gift enabling them to call him when the city is in danger; in 2005's Batman Begins, then-detective James Gordon creates his own signal light, inspired by an incident when Batman strapped the defeated mobster Carmine Falcone to a large searchlight, which created a roughly bat-like image from the light's beam due to Carmine's tattered coat.

In the comic's post-Crisis continuity, the signal was introduced after the Batman's first encounter against the Joker (not unlike the first movie) in Batman: The Man Who Laughs; in Batman and the Mad Monk, Gordon initially used a pager, but during a meeting with Batman he threw it away, saying that he couldn't sneak around in the shadows like Batman and wanted a more above-board means of contacting him.

On Batman: The Animated Series, it was introduced in the episode "The Cape and Cowl Conspiracy". On The Batman, Gordon invented it to summon Batman in "Night in the City" (somewhat similar to the 2005 film).

In the "Lovers and Madmen" story arc from Batman Confidential, which retells the origin of the Joker and his first encounter with Batman, Batman sees the Bat-Signal for the first time and assumes that Gordon created it to ask for his help in battling the Joker. When he reaches the rooftop, however, he finds that the Joker actually created it, and used it to force a confrontation with Batman.

Tim Burton film series
In Tim Burton's Batman, Batman gives the signal to the police as a gift so that when he is needed they can summon him.

In Batman Returns, Bruce has miniature Bat-Signals stationed up around Wayne Manor as a beacon to call him. It's only used when Commissioner Gordon needs his help when the Red Triangle Circus Gang attack Shreck plaza during Christmas.

Joel Schumacher film series
In Joel Schumacher's 1995 film Batman Forever, the criminal psychologist Dr. Chase Meridian uses the Bat-Signal to call Batman, to seduce him. Batman is slightly peeved at her usage of the Bat-Signal as a mere "beeper". A music video for "Kiss from a Rose", also from the film, features singer Seal performing the song while standing beside the Bat-Signal. Also in Batman Forever, The Riddler alters the Bat-Signal by projecting a question mark into the sky with the original bat symbol forming the dot at the base. Similarly, in Batman: Dark Victory, after brokering a tentative alliance with Batman, the Riddler changes the signal, projecting a question mark into the sky in order to let Batman know that he has an answer for him.

In the 1997 film Batman & Robin, Poison Ivy alters the Bat-Signal by changing the Bat-symbol to a Robin-symbol to lure Robin into a trap.

DC animated universe
In Batman: The Animated Series, Barbara Gordon uses it to contact Batman in "Heart of Steel" when she believes that an impostor has replaced her father. At this meeting, the signal is partially destroyed when Batman is attacked by a Harvey Bullock duplicate, and Barbara uses Batman's grapple gun to pull the robot into the signal, electrocuting it. Likewise, the real Harvey Bullock uses the signal for the first time when reluctantly asking for Batman's help in discovering who is trying to kill him in "A Bullet for Bullock".

In the movie based on the aforementioned series, Batman: Mask of the Phantasm, Batman is being hunted by the police on suspicions that he has murdered several gang lords, and Bullock, under orders from Councilman Arthur Reeves, tries to use the Bat-Signal to lure him in. Batman, knowing that it is a trap, does not respond.

In Batman Beyond, Terry McGinnis destroys the Bat-Signal after it is used for the first time in years by Paxton Powers, the son of Derek Powers (Blight), who uses the signal to contact Batman with the intention of using him to locate and deal with his father.

Christopher Nolan film series
In the 2005 film Batman Begins, Batman initially 'improvises' a Bat-Signal by draping crime boss Carmine Falcone over a spotlight at the docks after defeating him, his spread-out coat making the light look like a bat. At the conclusion of the movie, Lieutenant Gordon creates the actual Bat-Signal to call Batman to the roof for a talk, although he jokingly comments that he only uses it because they couldn't find any mob bosses to strap to it.

In the 2008 sequel, The Dark Knight, Gordon uses the Bat-Signal to remind Gotham of Batman's presence. The signal proves to be very effective, with drug dealers and criminals becoming apprehensive at its appearance. At the end of the film, after agreeing to blame Batman for the murders committed by Harvey Dent in order to preserve the former's image as Gotham's hero, Gordon destroys the signal in front of various members of the police force and the press.