Warning, there are spoilers for games post Gen II including ORAS and XY. SuMo temporarily fixed the issues here, but I doubt it will be preserved:
So, what is a Two Stage Evolution?
A two stage evolution is a Pokémon species that is the final stage of two stages within a evolutionary line. These species, unlike their three stage counterparts, tend to differ in a lot of areas, including:
-BST: Ignoring exceptions such as Snorlax, Archeops, Arcanine etc. BSTs of two stage Pokémon is generally much lower than three stage Pokémon, going around 470 and 500 BSTs.
-Stat distribution: Three stage Pokémon tend to have have evenly distributed stats which don't exceed each other radically. This because of the middle stage, which cannot drastically change from the base, but with its presence, the third stage cannot drastically change from it, either.
-Level of evolution: The level of a two stage evolution can vary wildly, but traditionally, it is generally below 30. In modern games, this limit has increased drastically, initially by some Hoenn Pokémon, and definitely in Unova Pokémon.
-Type: Three stage Pokémon traditionally keep the types they have for the entire route, or change it at one stage(generally the last, but Fire/Fighting starters and Togetic change at middle), and keep that changed type. Two stages also work similarly, but their type changes can be more drastic, changing their primary type(such as Steelix) which rarely happens in three stages(Fletchinder, for example).
-Moves: Second stage Pokémon, being available early on, generally do not have a lot of moves. Three stagers, on the hand, have a lot.
-Abilities: In three stage Pokémon, abilities generally don't change, in dual stages, abilities often change.
-Height/weight increase: In three stage Pokémon, height weight increase tend to be fairly balanced, but in dual stages, that's rare.
-Category: Similar to types, in three stages, categories are generally either kept, or changed at one of the stages. Species that change category in each step, such as Hydreigon, are rare. For dual stage Pokémon, there is generally a continuity but difference, though radical changes also occur.
So, what is a Middle Stage Evolution?
A middle stage evolution is the middle point in a three stage evolutionary line. These serve as the second stage of that line.
-BST: Ignoring cross-generational exceptions such as Rhydon and Porygon2, as well as certain exceptions that are meant to be good regardless of evolution such as Machoke and Doublade, most middle stage Pokémon are much weaker than their second stage counterparts, going around 370s and 410s.
-Stat distribution: Middle stage Pokémon tend to have more average stats than their second stage counterparts, which projects to their final third stages.
-Level of evolution: Middle stages tend to evolve very early, around 15-25.
-Type: Middle stages rarely change type upon evolution, and when they do, they tend to preserve that type to their evolution.
-Moves: Middle stage Pokémon can have some additional moves through Move Tutor and TM, but outside of that they tend to not differ much from their first stage, and those do serve similar purpose to their evolution.
-Abilities: Same as moves.
-Height/weight increase: Remains fairly balanced in all stages.
-Category: Surprisingly, category changes are not uncommon, but not common either.
So, why are these important, in-game?
Both of these groups are the common wild evolution groups you can catch in the wild without bothering training. However, they also serve the purpose of helping grinding in the wild, which is becoming more and more difficult since B2W2.
These evolutions also serve to increase the difficulty of late-game and post-game areas. Earlier games which had plenty of those used these stages extensively, but since Gen VI post-game areas no longer exist, making these more and more irrelevant.
So, why are these important, in-franchise?
If these stages no longer serve their original purpose, we start to see less of them. Each generation we see more and more standalone Pokémon, and less and less evolutions. Third stage Pokémon are now exceedingly rare, and two stage Pokémon are becoming rarer.
This means each generation can no longer support a large variety of powerful Pokémon, instead resorting to powerful gimmicks that will become irrelevant in the next generation.
This means more Pokémon go to the garbage bin, and this cycle just feeds itself.
So, why are these important, in-long term?
Overabundance of these gimmicky Pokémon instead of conventional evolutions also means that conventional evolution, as we know of it, is no longer relevant. This means this basic gameplay feature will instead be satisfied by other means, such as form changes, signature moves and Abilities and generational gimmicks.
As none of these features have long survival rate, this means the game will become more complicated, as it becomes less complex, with each Pokémon having some sort of a form to full in space, which in turn cuts the amount of actual Pokémon that is introduced.
Similarly, the fact that none of these Pokémon can actually hold against you in-game or in-competitive also means that the franchise itself will inevitably devolve into something where you are supposed to get your fan favorite Pokémon as early as possible and grind it as quickly as possible to break the game and play the metagame. And that just sounds yucky.