There is a lot of argument between the difference a CRT vs LCD makes, the difference between buffers, etc.. Additionally there has always been a lot of placebo effect going on involving small amounts of lag. This guide will not be covering how to make an ideal setup or any differences in lag between specific things. The only thing covered here is the effect of a small amount of lag (such as the 4ms that a nice LCD monitor has over a nice CRT monitor) actually has on gameplay. This guide will also be assuming that you are playing at a perfect 60FPS, with 0 frame drops/stuttering. That is an entirely different subject and not covered here. This guide will also not cover audio latency or auditory cues.
Before I start I'd like to mention that 0 lag is in fact impossible. This means that no matter what setup you are using you are playing with some amount of input lag. This means you are adjusting to the lag of whatever setup you are using, in effect, calibrating your muscle memory based on the success of your inputs. Even going between different CRTs (especially different refresh rates) will give you different input lag, meaning you have to adjust to the new input lag. You do this without noticing, partially because you probably didn't even know that you were at a different level of input lag. Brawl wifi players did this even with HDTVs with input lag upwards of 100ms and server latency starting at around the same. However, this guide will ignore this very real phenomenon and assume that it does not play a part, even though in reality it does, lessening the effects of lag even more.
First I want to cover what input lag actually means. When playing video games there is a cycle. Controller inputs and program variables are processed by the engine (your console, PC game, or Dolphin), then sent to your display (your TV or monitor), which then displays them in a certain amount of time (display lag), and the light travels to your eyes (insignificant time). Next your brain processes your sensory data into something it can actually use; this takes around 30% of your brain around 100ms to do. After that you process that image, deciphering what you are actually seeing, and then create a course of action, which you then carry out by sending electrical impulses through your nervous system into your muscles, which is a complicated but significant time delay. Once you do that and are doing an input on the controller, the game catches those inputs by polling the controller, which it does 120 times per second, that means that the console is sending an electric current through the controller and receiving a different current back depending on what state its various inputs are in. The cycle then begins over. Input lag is the time from when you make an input to the time that the result is displayed on screen. Display lag is the time from which the display receives the visual signal from the engine to the time the result is displayed on screen. Display lag is a subsection of input lag and will always be less than your total input lag.
So, now that we know what input lag is, what does it actually do? Input lag affects how quickly you respond to visual stimuli and can also affect your judgement of when game objects (such as Captain Falcon and the ledge) will collide. Some examples of things that input lag could affect are how quickly you dash towards Marth to punish him for dashing back, how quickly you shield when you see an unexpected attack coming your way, whether you just barely land a grab off a reaction tech chase, retaining maximum invincibility by letting go of the ledge as soon as you grab it, or L-cancelling after being launched into the air. What input lag doesn't affect is muscle memory. If an action that is muscle memory, it means that you already have decided when to do it based off of when you input a previous action, rather than when you see something on the screen. This bypasses the extremely inconsistent cycle described above allowing you to have much more precise timings on inputs. Because the entire input/response cycle is bypassed and you are deciding when to input an action based on a previous decision made, input lag does not play a part in muscle memory actions. You can tell if something you are very good at is muscle memory by trying to do it with your eyes closed. If you can, it is muscle memory. Additionally, since human reaction time rarely dips below 200ms, any action you perform automatically within 12 frames of another action is muscle memory. Some examples of muscle memory are wavedashing, ledgedashing, multishining, short hopping, SHFFLing, L-cancelling out of a jump, wavelanding onto a platform from below, wavelanding out of a short hop with Ganon, SHL and SHDL, moonwalking, or waveshine combos with Fox.
Hang in there, we are nearing the end of this guide, and it is now time to discuss visual stimulus reaction consistency. That is, the variation in time between complete rotations of the entire aforementioned cycle. This is composed of too many variable to actually track, including but not limited to: engine processing time, display lag, how quickly your brain processes the raw visual data, how quickly your brain processes the compiled sensory data, how quickly you create a course of action based on that data, how quickly your nervous system triggers your muscles (this and the previous three variables together are called reaction time), and how long after the input is pressed that the engine next polls the controller. We can however generalize these variables into three major groups: reaction time, input lag, and polling wait. Input lag includes everything from your adapter lag to engine lag to display lag and is more or less consistent on the same setup. However, reaction time is not. If you want to see the consistency of your reaction time, go to this website and use your elbow to take their test while playing Melee at the same time. Look at the extreme range of difference in your reaction times. This variance is added to your your visual stimulus reaction inconsistency. Because we do not see in frames or polling periods, when we press a button vs when the engine next polls the controller is completely random, resulting in a flat additional ±8.33ms to your visual stimulus reaction.
So then, what is the effect of this extreme inconsistency in your reaction to visual stimuli? It is humanly impossible to consistently react on the same frame after a visual cue (this is not the case with muscle memory). This is why things that rely purely on your reaction to visual stimuli are so incredibly difficult. As an example, if you are just standing next to Marth and he fsmashes you unexpectedly, you have over 20 frames to press shield in order to block it. That's over 300ms to complete the visual stimulus cycle once. However you'll notice if you try this in game, you will rarely if ever actually block the fsmash. For another example, we can take a look at pure reaction tech chasing. That means standing right next to where someone is about to tech, watching them tech, and then reacting as soon as possible to what option they take to punish it. This is very possible in TAS, that is, disregarding the entire visual stimulus reaction cycle. However despite significant proof that it is possible, the vast majority of players struggle to do it consistently in game, and even players that can do it mostly consistently still make mistakes. It is important to note that when players like DruggedFox successfully reaction tech chase, they are not always landing a hitbox on the last frame before the opponent is actionable. It's still just as inconsistent. Sometimes they grab and hit the opponent well before they are actionable. Other times they barely hit it. They are just reacting faster on average, which is why they still miss sometimes. Note that isolated visual stimuli are very different from tracking movement which would include things like pivot edgehogs, powershielding lasers, and timing ledgedashes by letting go of the edge as early as possible to retain maximum invincibility. Tracking movement is affected by input lag but not by reaction time.
At last the end has come. Finally, the implication of small amounts of input lag on gameplay. An increase in input lag results in a flat increase in the visual stimulus reaction cycle. Since this is so inconsistent to begin with, it will also be inconsistent afterwards. For example, if you have a total visual stimulus reaction time that ranges between 300 and 500ms, an increase of 10ms of input lag will result in a new range of 310 to 510ms. That means that if you needed to react to something within 400ms, you'd have a 45% chance instead of a 50% chance of reacting properly, disregarding polling rate. That means that 19/20 times the same exact thing will happen. Note that that 19/20 is heavily dependent on the example visual reaction time and will change significantly based on your own.
Now, here is an infographic showing the effect of 4.16ms of input lag (one buffer or approximately the difference between a high quality CRT and high quality LCD) on a frame perfect visual cue: