A Leap of Faith is a specific manifestation of the Behind the Black effect. In a Platform Game, falling off the bottom of the map will kill you, but in many situations there are places where you can still scroll down. If it is not possible to tell whether going down a hole will lead to instant death or a new area, it is a Leap of Faith.
If the character can jump horizontally far enough that his landing spot is not visible from the start of the jump, that too is a leap of faith if the platform doesn't scroll into visibility until after he's taken off.
A Leap Of Faith also occurs if there is a platform somewhere below the screen to land on, but there are still hazards around it. Thus, you end up having to guess where to position your character to land safely.
A careful level designer will ensure that any Leaps of Faith will be safe, or that they are always fatal. If the designer is inconsistent about this, it can be a particularly maddening form of Fake Difficulty. Alternately, an Easter Egg or other secret may be hidden behind a random Leap of Faith. If progressing through a level (or finding secrets) requires repeated experimentation with Leaps of Faith down otherwise indistinguishable pits, then you've got Trial-and-Error Gameplay.
If there are common, mostly useless powerups (like coins in Super Mario Bros. or rings in Sonic the Hedgehog), it's common to indicate pits where a Leap of Faith will be rewarded by arranging them in an arrow shape.
Compare Behind the Black, Trial-and-Error Gameplay, Fake Difficulty.
The Mega Man X games were fond of these. You could Wall Crawl (well, really, Wall Slide) and find that the screen would begin scrolling if there was something below. If it was a pit of instant death, it would not scroll and you could easily hop out.
The Mega Man Zero and Mega Man ZX series continue this trend, though ZX adds a long-range item scanner which removes the guesswork from the process when you use it.
The sixth game, though, had them purely as a function of bad level design. A hidden area beneath what looks like a bottomless pit? Good. Having to jump onto a moving platform (that you couldn't see) over instant death spikes (that you didn't know were there)? Bad.
Super Paper Mario hides clues indicating which pits will reward a Leap of Faith. Even if you jumped and were wrong, though, it only did 1 point of damage and teleported you back to the place you had just jumped from.
Mostly averted in the Sega Genesis-era 2D Sonic the Hedgehog games; almost any place where you can pitch yourself into space will either have a long landing strip, a tell-tale string of rings or a wall that stops you from going too far. Later 2D games — Sonic Rush is particularly egregious if you're not on the highest possible path — were less clever about this and it was very easy to die in a jump because there's No Way Out, or because you took the jump going the wrong speed.
A certain spiked pit in Mystic Cave, however, offers no chance of escape should you fall into it expecting another lower path. This is fatal, as there are no ways of escape, and the pit is too deep to leap out of, even if you had Super Sonic's improved jump height. Oh, and if you actually are Super Sonic, the spikes won't even kill you. Go grab a drink or something while your rings run out.
Not quite a leap, but the quicksand pits in Sandopolis Zone Act 1 are sinks of faith. Some lead to death, others to new parts of the level.
There's a level with this as its name in N. You leap off the platform and hit the wall, then wallslide down—-hitting the Trap Door switches (hidden behind gold) that create stairs and platforms for you to safely reach the exit.
The Empire Strikes Back game for the Game Boy consisted of nothing but this. With floor traps on virtually every level that were almost always one hit kills and no guide or maps available at the time, to beat the game required taking a never ending chain of leaps of faith until the player memorized the layout and learned when it was safe to jump into the abyss and when it would kill you. This is also an example of trial and error.
The Donkey Kong Country game for the SNES absolutely reveled in this; there is absolutely no way to find all of the hidden areas without systematically jumping, diving, and falling into every conceivable hole on the screen (of which there were many). Such an effort will take quite a while, but most of the secrets are based on patterns of details. e.g., when a barrel's sitting on the ground (as opposed to the floating DK Barrels), look for a nearby destructible wall or an enemy blocking a path. The sequel eased off on this a bit, thankfully.
In a couple of them, you can just see the top of a barrel, but not often. However, the port to the GBA made several of them more obvious, and removed a couple, such as the one at the start of the first minecart level.
Many of the earlier top-down Zelda games made it very hard to tell which pits led to a lower level and which dealt damage. You had to look carefully to see if the pits were totally black, or if they had a pattern in them representing the floor below. In more recent, 3-D, ones, not only is the bottom of pits almost always clearly visible, "bottomless" pits will often be over different parts of the same area.
And don't forget that loathsome dungeon in the side-scrolling Zelda II: The Adventure of Link. There is at least one dungeon in particular where, for once, falling in a pit not only won't kill you but is the only way to proceed.
The Legend of Zelda: Ocarina of Time had one Death Mountain ledge with either a Heart Piece or a Gold Skultula. If you moved the camera around you could see it, but you had to know to move the camera around.
Super Smash Bros. Brawl has only one example, which is still annoying. In the last part of the game, where they make you repeat almost everything, there are a few new areas. One of them is a wooden area based on a past one, which ends with two bottomless pits, only one of which will actually kill you. The other leads to a necessary battle. Since they are so close together, the map won't tell you if you're too far off from the door in question, and you scroll down with both. However, although it's incredibly obvious which one is the deadly one when it scrolls down, at that point you are too far down to do anything about it but struggle until losing a life. Luckily those lives can be quickly replenished.
In one area (The Path To The Ruins, if he recalls correctly) there is a series of what seem like narrow, bottomless pits, but one of them actually leads to a door hiding one of the item boxes necessary for 100% Completion.
Conquest Of The Crystal Palace has two such levels where falls don't kill you; instead, you get bumped back to a checkpoint near where the series of pits was. In one case, you can actually fall into a pit and discover an easy-to-defeat Bonus Boss that coughs up a one-of-a-kind Moon Mirror, which destroys all non-boss enemies (but It Only Works Once).
Bible Adventures, the David and Goltiath game, last level. The reason why AVGN couldn't beat it.
A slight tweak in Star Fox Adventures, in that the reward is not in the pit, but across a bridge of light that appears when you jump off the edge of this gap that's too big to jump across, in accordance with the instructions on a nearby sign.
Super Princess Peach handles this in an odd way. In early stages, there are no bottomless pits; falling into every hole is the only way to find all of the collectable items in each stage. And then later stages add the bottomless pits, punishing the player for using his conditioned reckless exploration skills.
Prince of Persia has two of these - the first with an off-screen platform, and the second with an invisible floor.
Lampshaded and parodied as early as E.V.O.: Search for Eden. Take a literal "leap of faith" off a certain cliff in a later level of the game and you get to transform into winged animals!
At one point in Last Ninja II, you have to walk up to a lake shore and jump onto an island on the lake. However, you have no way to know that the island is there or that you have to jump onto it (it's on another screen) - you see it only if you manage to make the jump. Worse, you have to jump from a specific spot - otherwise you'll immediately fall into the water (and subsequently die) without even seeing the island. The nearby swarm of killer bees which depletes your health at a very fast rate and is nearly impossible to avoid doesn't help much.
Quackshot, which cast Donald Duck in the role of Adventurer Archaeologist, had a section where you had to cross a vast canyon using invisible floating platforms which wouldn't appear until after you'd already jumped onto them.
Like the Donkey Kong example at the top of the page, many of the secret areas in Painkiller are guilty of this, specifically in levels where finding all the secret areas or their contents was a requirement for the 100% Completion. Sometimes you're at least giving a glimpse of the glowing holy items within and no immiadatley obvious way to reach them, but most times they're entirely hidden from view, meaning scaling every ledge and hopping over every cliff, or just look at a guide.
Portal has something like this in Test Chamber 18 - the large room with a spinning energy ball launcher is, apart from the side where you start, the other side, and the unreachable platforms with turrets on them, almost entirely a Bottomless Pit / Super Drowning Skills hybrid where "the floor here will kill you"- except for a small area next to the starting side, which you would never check to find at the beginning but becomes important at the end when you need to pull off a slingshot maneuver with the door wall you used to enter the room.
Pokémon Red and Blue have an example in the burned-out laboratory, although a nearby scientist will point out the jump to you.
Even without said scientist the player could be confident they won't get hurt, since NOTHING outside of battle can hurt you in the Pokémon games.
Either intentionally or due to poor design, many Super Mario World rom hacks will have you take a leap of faith, not knowing what lies below.
Kaizo Mario World does this, possibly in both the final castles with a door on a ledge, and Mario having to quickly jump off a falling Yoshi to hit invisible coin blocks to use to reach the goal, which is, of course, the infamous Kaizo Trap. This is among possible other examples which are probably too numerous to list.
Ōkami not only WANTS you to jump into what seems like a bottomless pit of flame, your sidekick actually ENCOURAGES you to do so.
Freeware game TAG: The Power of Paint averts this in the second-last puzzle. You have to make a jump that you can't immediately judge as possible - or at least, you couldn't if it weren't for the fact that the developers put a giant sign on the building you were jumping to saying "You Can Make It." Literally, in those words.
The original Rayman game has several of these, but they are generally indicated by minor powerups, and aided by the character's ability to look up and down.
The third episode of a shareware Platform GameSecret Agent. At one point, you're supposed to enter a teleporter, which will deposit you on a very short ledge with a robot walking on it. You're supposed to teleport just at the right moment, so that you can kill the robot (which becomes vulnerable to your attacks only every few seconds for a short time) immediately after teleporting (otherwise it will cause you Collision Damage and eventually kill you). The catch? You cannot see where the robot is before you enter the teleporter. (Even more infuriatingly, it's just a few pixels beyond the screen.) Therefore, it's purely a matter of luck (very good luck) whether you'll manage to kill the robot or die horribly.
Super Castlevania 4 also has a different variation of this: there's a pit right next to the stairs leading to Dracula and considering the overall difficulty of the game, most people won't give it a second thought especially since all other pits in the game just kill you. However this one has an invisible platform over most of it, and if you stand in the very corner of it, you'll be rewarded by a rain of hearts, a cross, a triple shot and a health restore in case you're missing any. However, notice the word "most" in the previous sentence: it's still very possible you'll walk off the invisible platform and fall into the actual pit while trying to backtrack.
In the game Eye of the Beholder II, there is a puzzle for which the clue is given of "faith is the key." In the puzzle, a sequence of 4 doors in a corridor must be opened by pulling four levers such that each lever opens a pit in front of the previous one, meaning that the fourth door-opener lever is inaccessible across a pit (albeit not a bottomless one, since the game doesn't have those). The trick, in this case, was to open the first 3 doors, walk down the partially opened corridor, and push a misaligned brick on the wall, at which point it becomes possible for the player to make a Leap of Faith and walk on top of the pit in order to pull the 4th lever and open the last door.
A puzzle sequence in the original Eye of the Beholder includes a part where you must jump down a pit in order to continue, an act that would normally injure everybody in the party. A nearby wall engraving reads "one leap of faith", and upon actually jumping down the pit, the party takes no damage.
Older 2D Metroid games featured similar moments in the form of Lava/Acid pools, where in later levels a path to a secret or the rest of the level is hidden behind a pool of normally hazardous acid that vanishes as soon as the player falls in. Fortunately, these pools aren't one-hit kills, so at worst jumping into a pit will take a few ticks off the health meter.
Spyro the Dragon had a couple such leaps. In the first game, for instance, the level Dry Canyon had a blind jump in order to reach (what was commonly) the last dragon; in the third game, the level Dino Mines had a jump very close to the beginning, another blind leap, that would take you to an egg carefully tucked away where no manipulation of the camera would allow you to see it, and the event's name is actually called "Leap of Faith". The Dry Canyon jump, at least, would just send you back to the start of the level; Dino Mines was a plummet to your doom.
Several hidden areas in Ratchet & Clank, usually leading to special bolts, would be hidden in places that seemed likely to kill you. Oftentimes they would if you weren't expressly trying to press Ratchet in the direction of a hidden tunnel or somesuch.
Ratchet & Clank Future: A Crack In Time adds a story-relevant leap of faith in Krell Canyon on Lumos. Alistair mentions that he and Ratchet's father enjoyed hoverbooting there as kids, and Kaden was never afraid to jump into the inky blackness (despite breaking his arm at least once). Naturally, as Kaden's son, Ratchet is expected to jump just as bravely.
In World of Warcraft, there is a mid level Horde quest where you must prove your strength of faith to the spirits through a series of tests. One of the tests is to jump from the highest point in the Thousand Needles, to show you have enough faith that the wind spirit will save you. You freefall down the side of a mountain, and just before you hit the ground, you are teleported back to the quest giver. While Death Is Cheap in World of Warcraft, especially to the players, you're not sure if you're going to be making a corpse run or not the first time you do the quest.
Another such leap quest was added in Wrath of the Lich King, where you're told by a gigantic water spirit to jump into the water, exposing yourself to her, as a show of faith. If you actually do it, she decides you passed and sends you back to the Kalu'ak for a reward.
With Cataclysm, the quest was removed, as the bottom of the chasm you had to jump in is now filled with Soft Water.
Also, Priests get a spell named Leap of Faith. It's used to pull party members to you.
At one point in Abe's Oddysee, you come to a sheer cliff overhanging what appears to be one of the many bottomless pits which litter the game. Chanting for a hint from the helpful fireflies will give you the message "Leap of Faith". It is clear what you have to do, although half-hearted players should note that doing a small jump or falling off the ledge will result in them being found to be not quite faithful enough.
One important area in An Untitled Story is accessible... by taking a several-screen tall leap from the peak of the nearby hill.
The game is full of them. In fact, this is the only way to leave the first screen.
Kirby 64 has a crystal shard hidden out of view between two high cliffs on Neo Star. It's marked by stars, though.
Final Fantasy VI - At one point in the plot, you run out of places to go except down a large waterfall. Your options are "Jump!" and "You crazy?".
BUG! has a truckload of these, especially secret areas (Reptilia comes to mind). Want to find a secret area? Just jump off into the foreground, and "hopefully" land on a platform leading to the secret area!
Some items in Dark Souls require this. There's even an item, the Prism Stone, to determine the lethality of falls. Another example in Dark Souls is entering the Abyss, a black void zone used for a boss battle. It requires a special ring to enter, as well as a Leap of Faith.
The Legend of Zelda: Skyward Sword features a moment in which you must jump off a ledge into what seems like a lava pit. When you're falling, a platform appears just above said lava pit. However, the game does tell you that you have to leap off the edge.
Indiana Jones and the Last Crusade. The platform itself is only visible from a very specific angle; when seen directly from Indiana's perspective while walking on it, it is invisible, and Indy has to spread dust on the walkway to make it visible to anyone following him.
The LucasArts computer game version had the same leap of faith test; in the game, you passed if you didn't hesitate before crossing. If you waited too long, any later attempt simply resulted in a fall to your death.
In The Name of the Wind at Elodin's command Kvothe steps off a roof to prove his trust in the power of the Master Namer and become worthy of studying Naming under him, explicitly explaining that Elodin required a leap of faith. He breaks three ribs and dislocates his shoulder, and Elodin refuses to take him as a pupil on the basis that anyone stupid enough to jump off a roof because his master told him to has no business studying anything so dangerous as Naming.
In Uglies, Tally's journey instructions to "at the second [break], make the worst mistake", means to drive her maglev hoverboard off a broken railroad bridge, upon which the board activates from the metal deposit under the river, enabling her to continue her journey.
In The Diamond Age, there's an entire community of people who all implicitly trust one another, because every person who petitions to join them must perform a Leap of Faith. For example, one person is told to wait at the top of a cliff at a specific time wearing a bungee harness, with the other end of the bungee cord not tied to anything. Ten minutes later, they must jump. They have to have faith that a member of the community fulfilled their order to go to the bottom of the cliff and anchor the cord. It's not always a literal leap, though; another ritual is described in which the applicant leaves a loaded gun in an empty room, then comes back in, points it at their head, and pulls the trigger, trusting that their "sponsor" came in and emptied the gun in the meantime.
In My Little Pony: Friendship Is Magic, when stuck between a long drop onto jagged rocks and a hungry, furious hydra, Twilight Sparkle panics. Pinkie Pie uses these exact words to coax her friend to make the jump. She actually misses the ledge, but a well-timed swamp bubble saves Twilight from drowning.
In the Visual Cliff experiment, researchers placed babies and mothers on either side of two small, raised platforms. The mother then began calling for the baby. The trick was that there was a clear plastic floor over the gap, and the baby could quite safely crawl across it to the other side. The experiment was done to find out how soon we develop depth perception (and the instinct to not make leaps of faith).
Most of the babies (of several species) refused to crawl off the cliff. Human babies would generally do it until they reached a certain age, or if the mother acted frightened of the gap. Baby turtles would attempt to dive into the glass, confusing the cliff with a pool of water.
In the Grand Canyon, there is a skywalk made of 2 inch thick glass (they drove a semi across it to show its durability). The Native American tribe who runs it keeps the skywalk polished so the center of the walkway appears nearly invisible.
One example was in the Menlyn Mall in Pretoria, South Africa (the largest mall in Africa and largest enclosed mall in the Southern Hemisphere). It's under a roof but the main hall has an open plan, with higher levels only having floors around storefronts so that you can see the roof from the ground floor. This means that on higher levels, you need to cross footbridges to get to a store on the opposite side of the same concourse. Near one entrance, there was a specific bridge made of clear glass. People refused to cross it, preferring instead to go around the entire hall... so eventually it was sandblasted, and people now use it without a second thought. (Obviously, women wearing dresses had other reasons not to cross it, but still.)