First off, cool story. If your looking to write it all out here is a great spot. Now I'll explain tiers to you.
What are tiers?
Before I explain tiers I'm gonna explain Pokemon Showdown! (or PS! for short). This is an online battle Simulator for Pokemon, and to the best of my knowledge, the most used battle simulator.
They have a few different rules than game freak on PS! One of which is tiers. They take how much any Pokemon is used and place it in a tier. You build a team using your tier down and battle other teams only in that tier.
The tiers aren't based off of stats, or legends or typing, but how much people use it. An example of this is Mega Mawile, it's base stat is only 480, but it was used so much that it was placed in the highest tier, but Mega Metagross is a tier down from mega mawile, even though it's base stats are 700.
Here is a link to PS! if you want to check it out. You will find better battlers here, it has more capabilities and it doesn't require hours of breeding and EV training.
What do they represent?
I said this somewhat above, but I will expand on the tiers here (telling you what the tiers are).
The top tier is ubers. Every Pokemon is allowed in ubers except mega rayquaza.
The next tier is OU. Every Pokemon is allowed here, except uber ranked Pokemon.
The next is UU, every Pokemon that isn't in ubers and OU are allowed.
Then it's RU, then NU, then PU, then NFE then LC.
One thing I would like to point out here is that PS! has the same capabilities as normal battling. You can't give slaking wonder guard and sweep everyone.
How do they influence your way of playing?
Well, I said some of the pluses of PS! above, but another that would benefit you is it's completely free. Your battling will improve dramatically and (to me anyway) it's more fun than playing in game.
Are they Nintendo's rules?
No, you don't need to follow these rules if your playing in game, but they have better rules for the most part. Some of the rules are:
Sleep Clause (only one Pokemon can me put to sleep at a time)
Item Clause (Any number of one item on a team)
Swagger Clause (Swagger is banned)
Moody Clause (Moody is banned)
Species Clause (Only one of each Pokemon is allowed)
There are more rules depending on what tier you play, but these are the main ones.
How do they impact Pokemon?
It makes Pokemon more of a skill game versus a get the best Pokemon game. You all have the same options of Pokemon and you won't be restricted by time, money or anything else to obtain the Pokemon.
Why should you care?
Well, I've made multiple points throughout this entire description of why you should care. I didn't understand PS! at all, but I went on and picked it up rather quickly, then I had sempiternus tutor me and got decent at battling on there. It takes less time to obtain the Pokemon and more time just playing the fun of the game.
EDIT: Here is a link about IV's in case you don't know what they are.
Also found this that brings up some good points on tiers.
Found this on what Pokemon are in certain tiers in case your curious
Sorry for all the links :P
Sorry for the length of this, but I wanted to make sure you understand. If you don't understand anything or have any questions please ask!
Hope this helps!