My regular listens:
My Brother, My Brother, and Me - My favorite podcast. Dear Abby for the modern era. 3 brothers from West Virginia offer advice to Yahoo Answers questions and viewer submitted questions. Every episode is consistantly hilarious. Even their ads are usually great.
Jordan, Jessie, Go! - Another hilarious one. I don't even really know how to describe what makes it great. It is a talk show with 2 college friends chatting with funny people. Just try it.
The Adventure Zone - A Dungeons and Dragons podcast from the brothers from My Brother, My Brother, and Me and their Dad. This one just started
Judge John Hodgeman - Basically a comedy podcast version of Judge Judy starring demicelebrity John Hodgman.
This American Life - I think most people know this one. Usually true stories about everyday people set around a theme.
Backstory - An American History podcast from 3 History professors. They usually examine a single topic from the point of view of the 18th century, then the 19th and then the 20th.
Hardcore History - Dan Carlin's extremely in depth history podcast. He is covering World War I right now and is already over 10 hours in.
Common Sense - Political podcast with Dan Carlin from Hardcore History. I don't always agree with him, but he is smart and usually sees things from an angle no one else is talking about.
The B.S. Report - Bill Simmons. You probably know this one.
The Indoor Kids - Comedians Kumail Nanjiani and Emily Gordon discuss video games with a guest - usually another comedian or a game developer.
The X-Files Files - An X-Files rewatch podcast with Kumail Nanjiani from The Indoor Kids.
Idle Thumbs - A video game podcast that comes at gaming from the dev's perspective, the fans perspective, and the press perspective. Mostly absurd but they can get into some pretty deep conversations about gaming. One of them is a Polygon journalist, several of them are the guys that led development of the first Telltale Walking Dead game.
The Jeff Rubin Jeff Rubin Show - Jeff Rubin from College Humor interviews interesting people, e.g. A guy that reviews pens, a video game speedrunner, one of the brothers that created Homestar Runner