There are some champions which are commonly known to "fall off late game", such as Lee Sin. But what is it exactly, what defines if a Champion "falls off" or not? Or to be more precise, what (for example) makes Lee Sin less useful compared to other champs in Lategame?
I think the best example of someone that falls off late game is AP Sion. Early game he is an absolute terror and can destroy just about anyone; late game he struggles with relevance.
@Mario's answer suggests that this is due to poor scaling, so lets take a look at Sion's abilities and base stats.
A .9 ratio is nothing to scoff at. For comparison, Karthus' Lay Waste (single target) and Requiem have a .6 ratio and his defile is only at .25.
Let's compare sion's base stats to mordekaiser's.
As you can see their level 1 base stats are extremely similar, assuming they're fighting an AP base stats would tend to suggest that Mordekaiser would be stronger early. At 18, Sion has 390 additional base health over Mordekaiser - suggesting that Sion would be stronger than Morde late game. This, however, is clearly not the case.
So then what exactly makes Sion fall off late game? His kit. Sion is extremely good early because he can effectively stun and shield burst to kill most champions. Once late game hits champions will generally (either naturally or build to be) tankier and be able to withstand this combo and peeling becomes much more effective. (ie, the ADC can lifesteal a respectable amount during CC)
Ultimately, the key to any champion falling off towards late game is their inability to no longer burst a target down and their impact in team fights is greatly reduced because of it. This seems to be the case to me with any early game champ (shaco, lee sin, renekton, etc).
(stats taken from league of legends wiki)
There's no general rule of thumb for this, however consider it a bit like a carry's power curve - just reversed. The Champion won't become weaker, but his power increases less compared to other Champions on the field.
I'm not that good at number crunching with LoL mechanics, so I'll make up two abilities with different AP ratios:
In this example, the second ability will provide you with a stronger head start, while the first ability has the far better damage increase with more ability power. So in this example, the second ability would get weaker in comparison the more ability power the characters have got.
But this isn't just about ability power ratios or damage values. Another example would be fixed reductions to armor and/or magic penetration: Early on these might take 50% or more of a character's defenses, so they're rather strong. However, later on the enemies might aquire more resists and therefore cause the effective reduction to become smaller and smaller (it "falls off").
So, I'd try to explain it this way: If a Champion's numbers are static, i.e. fixed values, then he'll most likely be weaker the longer the game takes compared to Champions having dynamic numbers. However the Champion with fixed numbers will have a far stronger early game, so it also depends a bit on how long you expect the round to be. If some ability has both, i.e. a fixed portion and a dynamic bonus (the usual case), then it really boils down on how these values compare to the enemy. There's no general "good" or "bad" as it always depends on the enemies and the game length as well.
I agree with Mario's answer. But i'd like to add that some champions due to their mechanics are more suitable for early game playstyle, which means that they are stronger in small teamfights or 1v1 and they are weaker in teamfights. Usually those champions are melee fighters like Lee Sin in our case.
Another factor is the late game power of ad carries. Champions like Lee Sin can be really strong against ad carries early game because they are tankier by default and the ad carries don´t have enough damage to kite them effectively. Late game though the carries scale in damage and their ability to kite becomes much higher.
So late game you end up having a champion that can be kited easily from a distance without any real cc or much aoe to contribute to teamfights that become much more frequent. So you get the feeling that the champion "falls off" even though his abilities may remain strong in smaller fights...