The move consists of Captain Falcon throwing a powerful punch, with his fist engulfed in flames shaped like a falcon. It is one of the most powerful moves in all three Super Smash Bros. games, with the highest knockback of any of Falcon's attacks (including charged smashes). This move is one of the few attacks in the game that is more powerful than the Home-Run Bat at high damages (and since Captain Falcon's home-run attack has an upward trajectory in Melee, it is always more useful), and thus players use it in Captain Falcon's world record Home-Run Contest strategies. The Falcon Punch is very slow, with a start-up time of 52 frames in Melee. It is also very noisy, especially since Captain Falcon has to announce it with a "Falcon...PUNCH!" every time he uses it, making it even more predictable.
The move is one of Captain Falcon's primary KO moves in Super Smash Bros. It is best used when timed to match an opponent's recovery, KOing them as they recover.
In Super Smash Bros. Brawl, Captain Falcon can turn around during the start-up of the punch, increasing its damage. This is a Reverse Falcon Punch. It also has 42 frames in this game. The Reverse Falcon Punch does slightly more damage than a standard one, but has slightly more start-up time. Although the move can be reversed in Melee like all neutral special moves, no changes are noticeable bar direction. The move has become quite a meme around the internet, so much that it is even referenced in Snake's codec conversation.
Falcon Punch has four different stages of knockback and damage which is determined by what type of punch is performed.
Description from the Melee instruction booklet
Release a fiery blow after a brief charge.
The move originated in the Super Smash Bros. series and is generally considered part of the SSB universe. However, after SSB's release, Captain Falcon famously performed a Falcon Punch in the F-Zero animated series, F-Zero: Falcon Legend (adapted to Western markets as F-Zero: GP Legend), where he uses it to defeat Black Shadow. (See video.) It has become quite memetic, as the clip makes it appear that the Falcon Punch causes an explosion that can be seen from space, when in reality, Captain Falcon is merely using the Falcon Punch to prevent Black Shadow from escaping said explosion. The meme also involves Falcon Punch imitations of some sort, with voice clips inserted (though special effects are not necessary). This is reinforced in a Subspace Emissary cutscene, when Captain Falcon performs a Falcon Punch on a giant R.O.B. in a very similar fashion to the clip from the animated series. The meme has crossed all the way back to Super Smash Bros. and is referenced in Captain Falcon's codec with Snake.
As a testament to the move's popularity, it seems to have come full circle and crossed back over into F-Zero canon: it is mentioned in F-Zero GX's ending song, specifically in the line "When a mighty illusion threatens your lunch / Falcon will be unleashing the Falcon Punch."
The Falcon Punch in Super Smash Bros. is easier to combo with than in Melee and Brawl due to it having only SDI and much less start-up and ending lag than in the subsequent games. It is generally used as a combo finisher; common tactics include throws forward throw -> Falcon Punch and up aerial -> Falcon Punch. It can also be used for recovery when Falcon is recovering high, as it gives additional horizontal distance as well as minor vertical distance when angled up.
The Tapion Punch combo, named after Tapion, is a two-hit combo that leads into Falcon Punch after an L-cancelled down aerial. It works on Fox as low as 75% damage and another Captain Falcon as low as 80%. It can be avoided with DI.
Another combo is the G-Regulate combo. Named after G-Regulate, this combo works similarly to the Tapion Punch Combo, but uses a platform. Captain Falcon must L-cancel a down aerial onto a platform so that it hits an enemy below him. Against some characters at some damages, Falcon can combo his down aerial into Falcon Punch. It can be avoided with DI.
In Melee, the Falcon Punch was made slower, though the punch itself seems to be stronger and now has a full 3D model, complete with better fire effects. Due to Captain Falcon's down aerial being much slower than in SSB64, the Tapion combo was made more unreliable, but with L-Canceling, it's still possible.
One of the more famous combos is the Sacred Combo. Originating in Melee, it is a combo that uses Captain Falcon's Forward Aerial and the Falcon Punch. It is very hard to perform. The combo itself is performed by knocking an opponent off the stage with Captain Falcon's forward aerial (the Knee Smash) and then jumping off the edge of the stage before striking the opponent with Falcon Punch, in mid-air. If performed successfully, this combo results in a KO in almost all cases, although by performing this combo, Captain Falcon may fail to recover back to the stage in Melee if he jumps too far off the edge to perform his Falcon Punch. In either cases, KOing an opponent with the Sacred Combo is widely regarded as being one of the most satisfying ways to do so.
In Brawl, the punch appears to be faster than its Melee counterpart, but nowhere near as fast as it was in 64. Due to the physics changes, the punch is weaker, but due to the new Reverse Falcon Punch gimmick being more pronounced than in Melee, Falcon gains a new variety of tricks with it. The attack also yet again, sports new effects and improved animation.
One combo in Brawl is to double-jump facing away from the opponent and right before he falls, reverse Falcon Punch. The character can then tilt the motion towards the opponent to reduce the time it takes to charge on the ground, strengthen the punch by reversing it, and use the attack just as he hits the ground. Best to use it on off-guard opponents that are expecting the player to flee.
One can also use it while running then RAR the Falcon Punch. In order to this one must run and tilt the control stick/d-pad back and jump at the same time. They'll jump while keeping their momentum forward, during this time, usually at the start of the jump, start the Falcon Punch and reverse it. Right when they land, they'll punch anything in front of the player. This technique also allows for more control over the distance traveled with the Punch. For instance, when a normal Falcon Punch is done from a jump, the distance traveled depends on how high the jump was. In contrast the "RAR'd" Falcon Punch allows one to basically, move either further (because of the forward momentum made if the forward direction is held) or even slightly behind (if the stick is tilted backwards after the punch is reversed in the air, the momentum will pull Captain Falcon slightly backwards). This may take quite some practice to master.
The above method can also be applied when hanging on an edge, albeit much more dangerous because of the likelihood of an SD. When hanging from the edge, one must jump, then tilt away from the edge (this will not visibly make Captain Falcon turn, but he will be turned around in the game's registering), then reverse Falcon Punch. When done at the tip of the edge jump, the punch will be thrown a little bit off the ground, so if one wants the punch to activate on the ground, then start the Reverse Falcon Punch a little after the tip of the jump. The hardest parts are making sure the game still registers Captain Falcon as facing away when performing the reverse, and also making sure one has stopped tilting away from the edge when starting the Reverse Falcon Punch. This method is slightly more difficult and a lot more frustrating when done incorrectly than the normal "RAR'd" Falcon Punch, but it can catch potential edgeguarders off guard.
If there's a fray happening without him involved, a good strategy is for the player to throw in a Falcon Punch, and it will have a good chance of hitting at least one person. To do this, Captain Falcon will have to double-jump at the edge of the fray, not the middle, or the punch will most likely be interrupted. If done right, he can hit all of the opponents in the fray, off the edge.
The following are three videos showing the Sacred Combo being performed: