Roguelites/likes are games based on the classic video game Rogue which was a dungeon crawler that reset your progress every time you died. Every run would reset everything about your character, but as you progressed through the run, you would become stronger.

Rogue-likes are games that are similar to Rogue. Rogue-lites are games which are also similar to Rogue, however instead of completely resetting all your progress when your run ends, you have some materials/resources/skills/abilities that are leveled up outside of runs. Smelly nerds care about the difference, I honestly don't really care unless it scratches my Roguelike itch. I usually just call them all roguelikes.

Run

A "run" is considered the main gameplay loop of a Roguelike. You start, you play, you level up, you get stronger, you die, you reset, you start again.