I'm a pretty big podcast junkie so I'll also post a couple of my faves!
One of the original bad movie podcasts, if not the original. The hosts are three NY based best friends who watch and rip on movies that are either commercial or critical flops. Near unstoppable and hilarious tangents every episode. Plus it's just fun to listen to host bounce off each other. One's a writer at the Daily Show, one the new head writer of the new MST3K, and the other owns a bar. They've got amazing chemistry.
My Brother, My Brother, and Me
Three brothers dole out advice by either answering emails from listeners, or selecting any number of inane Yahoo Answers questions. These guys are hilarious and are just flat out goofballs with hearts of gold. Lin Manuel-Miranda of Hamilton fame also is a big fan apparently.
The Adventure Zone
The above mentioned brothers, plus their dad, playing D&D. Equal amounts of hilarity, but also following a pretty stellar fantasy narrative. This one's worth listening to from the very beginning.
This one's a little harder to describe. Dan Harmon, (Creator of Community, Rick & Morty, other things) hosts a weekly show where he talks about... things. About life, current events, Hollywood culture, writing, racism, growing old, whatever. He goes in front of an audience and improvises (or justs never prepares) for his show. It's a lot of him working out stuff, which is fascinating because he's a legit genius. Plus he will occasionally break out in shitty freestyle rap, and randomly have regular guests like Kumail Nanjiani, Steve Aegee, Curtis Armstrong. Also his co-host Bryan Davis (Whose Line is it Anyway) is just as hilarious and quick witted as Dan is neurotic and clever. It's not for everyone, but this podcast has some amazing moments.
Cool Games Inc.
Two guys from Polygon.com "start" their own game development company, and take suggestions from twitter to make silly games. They'll concoct silly and dumb game systems for games simply based on titles like "Tim McGraw's What if? Truck: Fates" or "DreamWorks Kissin' Party." Fun stuff.
Matt Besser and 4 other improv comics from the UCB theater get together to do some audio improv. real "theater of the mind" stuff. As someone who does improv myself, I like to listed to this to get a taste of how the pros do it.