The Gentleman is a nickname that refers to the third hit of Captain Falcon's standing neutral attack combo, when it is not followed by rapid punches. It involves Captain Falcon kneeing his opponent, with low to moderate knockback at higher percents. It deals 10% damage in Smash 64, 8% in Melee, and 5% in Brawl and Smash 4.
In Japanese, its official name is the Knee Butt (ニーバット).
The mechanics of neutral attack continuations in Melee are complex. In essence, a neutral attack's infinite is triggered if the player presses or releases the A button a set number of times while executing the first few hits, but these "toggles" do not count if the character is in freeze frames, or is standing still after a hit yet still within the window to continue into the next hit (or "idle continuable time"). Captain Falcon specifically has a value of 4, meaning that under normal circumstances the infinite launches after 4 toggles, or only 2 full A-button presses. Since the player cannot get to the Gentleman without pressing the button 3 times, avoiding the infinite requires "dropping" 3 of the 6 toggles by placing them inside freeze frames or idle continuable time.
With such technical information unavailable for a large part of the technique's life, players created several subjective ways to achieve it. The player can perform the third neutral attack input by pressing any button in conjunction with the neutral attack (Isai finds this easiest to achieve with Z). S2J discovered that one can hold down on the control stick after the initial jab and continue the rest of the combo to do the Gentleman. The safest and most guaranteed (but not necessarily the easiest) method is to simply allow each jab to complete its animation. However, the first two methods do not work if the jab combo whiffs (even the last method has a chance of failing if the first or second jabs whiff), leaving Captain Falcon wide open if the opponent can escape it in time.
In the PAL version of Melee, Captain Falcon's infinite trigger value is changed from 4 to 7, so it cannot be triggered until the player presses A a minimum of 4 times (and none of the releases are dropped). This makes the Gentleman trivial to perform, making the combo much safer on whiff and on hit. This also applies in Smash 4, where one can even simply hold the jab button to get the Gentleman out of the jab combo; in fact, the rapid jabs and Gentleman act as different, completely separate enders to Captain Falcon's jab combo, as mashing the attack button through the combo simply skips the Gentleman altogether (this is also the only way to get the rapid jabs, since they do not activate on whiff unlike the Gentleman).
In Melee, the Gentleman is often considered to be the superior ending to Captain Falcon's neutral attack combo, as it has little ending lag and usually knocks opponents away far enough for it to be safer than the rapid jabs that normally follow it. This also allows for much better followups at higher percents, potentially setting up an edgeguard situation or even for the Knee Smash where the rapid jabs would render Captain Falcon unable to do so otherwise.
In Brawl, the Gentleman has severely nerfed knockback for the purpose of reliably linking to the jabs. Because the knockback has been lowered to the point where it cannot knock opponents back a significant distance until 200% or KO even at sudden death percentages, cancelling the jabs for the Gentleman is nearly pointless, except against characters without quick aerial moves, in which case the Gentleman can be mixed up with another gentleman or a jab reset.
In Smash 4, the Gentleman has reverted almost entirely to its Melee iteration, having increased knockback to set up combos. As mentioned above, it is much easier to perform, as one only needs to hit the jab button three times or even simply hold down the attack to execute it. If one mashes the attack button, however, Captain Falcon will skip the knee entirely and enter his rapid jab combo, which now has a knockback finisher of its own that makes it almost just as safe regardless.
In Ultimate, Gentleman itself has not changed at all since Smash 4. However, Captain Falcon's neutral attack as a whole has been changed, with it connecting more reliably through changes to the other hits. It remains as easy to perform as it was previously, as does his rapid jab.
According to Toph, this technique was named after a Japanese smasher with the tag "gentleman". He and Isai placed a bet where the player who hit the tech most frequently would have the move named after them.