Match Making Rating. It defines who you play against.
Now they community of league of legends likes to use the word Elo. But in fact they are using that term incorrectly.
Elo is the MMR used in chess and a few Online 1v1 games. Riot used the standard Elo system for a year or 2 back in the early days of league of legends. but quickly made changes to the Elo system to make it work better in the league of legends environment. This is what is known as MMR.
MMR is kind of an arbitrary number and none of us really know what the parameters are. Most likely its somwhere around 1-10,000.
The concept behind this is that lets say you have 1,500 MMR. You theoretically would be matched with and against players around 1,500 MMR. They could be also 1,500 or they could be at 2,000. We dont know what the allowed gap between players is. we do know that the range increases based on time spent in que. Now this also applies for the other players. For example. A diamond player with 8,500 MMR might have a 3 min que where they are playing against all diamonds with around 8,500 MMR. But in another 3 min que they could be matched against a challenger player with 10,000 MMR because that challenger player was in que for 15 min and so his range dipped into the diamond players and the game threw him in there with them.
So the Original Poster of this discussion is using the wrong words. He is trying to say that rank has no bearing on the MMR and he is somewhat correct. Rank is a rough indicator of your MMR. but it isnt the be all end all.
So the PSA part of this is. Elo and MMR are the same thing.