Broken means the champion design breaks some aspect of the game if said champion is viable.
1. Support Carries - This is problematic because these champions do not need to compete for gold to deal massive damage. This breaks the competituve nature of the game. Why should the enemy Brand be able to 1 shot my entire team with his high base damage + passive despite having a nominal amount of gold and experience? It feels incredibly unfair and unfun to play against. It especially sucks that this type of champion often does the most damage in the entire game, despite having the least resources in many cases.
2. Invisible Assassins - This one is just obvious: no one wants to be 1-shot from invisibility. For squishy champions, there is no reasonable counter play to an invisible assassin. Some people argue that the fact that rhese champions reveal themselves when they're near you means that they aren't broken, but this is clearly a bad argument. If the assassin has gotten close enough to your adc to be revealed, it is already too late, lol.
3. Scaling champions with lane dominance - Think Kai'sa, Gangplank and increasingly, Vayne. These champions are characterized by having a nigh infallible lane phase, while also having the ability to out-scale most champions in the game. If the champion doesn't outright win most matchups, they at the very least have a very safe lane phase that allows then to circumvent the stage of the game where the enemy team is supposed to have the oppertunity to shut them down.
I typed this on my phone, so I apologize if my grammar sucks because I am not going to read this over.