Solo Queue is when every player on the team is... well... solo. Not playing as a group with any other player. Essentially is five random people fighting five random people, all dealing with the same communication issues that causes and all forced to use pings and play with that lack of easy coordination in mind.
Dynamic Queue is a mode that lets people play with friends they are grouped up with... often in voice chat with each other. These teams have superior coordination and don't have to worry as much about not knowing what their allies are planning or doing and they react better as a team for the most part. They're also usually less prone to bitching and fighting with each other since they're all... friends. Or at least, would sound like jerkbags over voice chat.
People's issues with the modes are varied. MY issue is that Dynamic Queue is just flat out unfair to the solo players on each team. Riot insists that premades almost always get matched AGAINST premades... but that still means there is one or two players on each team who are out of the loop on team coordination and struggling with the usual problems one has in multiplayer games of this nature... but they are the ONLY ones who have this problem. Thus... they are at a disadvantage right out of the gate in a game that is supposed to be an even playing field for everyone to start. There is NO advantage to being a solo player in dynamic queue. Any game where a premade shows up, the solo player is at a disadvantage compared to them... even when he has a premade on his team as well. He still has to struggle with the lack of communication with his team... AND he's likely against a premade and thus a good portion of the enemy is far more coordinated with their teammates than he is with his. It's set against him no matter what. Not an insurmountable thing to overcome... but imagine if the new game mode made solo players earn say... 2 less gold per minute in one game out of every ten. Or they were unable to hear pings or something. You could win with that handicap... but why does that handicap need to be there?
And that is why people are upset. Because the premades are being favored by the new game mode and Riot is essentially saying that they prefer these premades get to play with each other than that the games be as fair and balanced as they can be. AND the imbalance is always against the solos. Shockingly, solo players are pissed.