I'm a little embarrassed to say that I've heard "roguelike" mentioned frequently, but I still have no idea what it means. What are some popular roguelike games that I might have played or could easily play to get an idea?
Brogue is an excellent and free roguelike. You can grab the Community Edition of it over here: https://github.com/tmewett/BrogueCE
It's fun to get a group of friends together and play a daily seed run... see who gets the most gold, who gets deepest, etc.