What determines an early/mid/late game champ?

It depends on two things:

1. Item scaling - champions that scale very well with items tend to need a lot of time to get strong, but are stronger than most people after that time. Champions that aren't item-reliant can do a lot more early but won't improve much as the game goes on. Champions like have poor early games and strong late games because their kits mean they need tons of stats to function.

2. Type of fight the champion thrives in - early game has a lot of duels and small skirmishes, whereas late game tends to have a huge amount of team fights. Champions that are strong in small fights tend to be very strong early, but then they struggle quite a bit later as it's hard for them to use their kit properly. is a great example of this - while she has pretty good AP scaling and her passive seems to encourage her late game, her kit is much stronger in 1v1s and small fights than it is in team fights, so she generally falls off after laning phase.

There are several reasons I am sure, but one is base stats vs scaling. A champ that gets a bigger damage boost from ranking a skill up favors early game, where like a mage that does more damage per ap than others is stronger late game.

Also champs like Nasus and Veiger do more damage the more creeps they kill. and some champions just don't bring as much to or do as well in full on team fights.

Sorry if that is not very clear.

How well their stats and kit allow them to transfer to different parts of the game.

Lee Sin is great early because he can out damage pretty much anyone without items. He is bad late however because if you dont go tank or have insec skills, you will die instantly and help the other team more then your own.

Someone like vayne is godlike lat game because having about 3 items and boots allows her to use the full extent of her kit and out dps most other champions.

Itemization, kits, and counters are normally what determines what part of the game champions fit in.

Good base damage --> Early/Mid Good ability scalings --> Mid/Late Many steroids (ADCs) --> Late (they increase your DPS a lot) AoE Damage --> Mid/Late High Range --> Mostly Early High CD reduction per rank --> rather Late DPS --> Late Burst --> Early