Podcasts I've been enjoying listening to recently include the following (I have not made my way through the back catalog of all of these, by the way):

• "Politically Reactive" with W. Kamau Bell and Hari Kondabolu: funny interviews about current political events. A favorite episode: their interview with Rachel Maddow.
• "How I Built This" from NPR, focusing on how people built new endeavors. A favorite episode: The story of Jane Wurwand, founder of Dermalogica.
• "Our Debut Album", in which comedians try to write songs. A favorite episode: "King of the Deep".
• "Say Why to Drugs" with Dr. Suzi Gage and Scroobius Pip, in which they share the current research on the effects of various drugs. (The bit of the intro where Pip says "pro-truth and anti-myth" has extra reverb or something and I adore it.) A favorite episode: caffeine.
• "Wolf 359", a scripted scifi podcast about a spaceship where things are getting sketchy. Listen from the first episode, but note that things get much more Battlestar Galactica-y starting in season 2.
• "The History of Philosophy in India", by Jonardon Ganeri and Peter Adamson. A couple favorite episodes: on the Mahabharata, and on women philosophers in ancient India.
• "Song Exploder", by Hrishikesh Hirway -- in each episode, musicians talk about how they made a particular song or score, and you get to hear the early components and drafts, and then you hear the finished piece. A few favorite episodes: Garbage ("Felt"), Weezer ("Summer Elaine and Drunk Dori").
• The Recompiler podcast, by Christie Koehler, on current tech events and interviewing interesting people in tech. A favorite episode: interview with Heidi Waterhouse.
• "Track Changes", from Postlight (a consultancy in NYC), in which Paul Ford and Rich Ziade talk with each other and/or guests about tech and design stuff. A couple favorite episodes: parts I and II of an interview with a guy who was a big cheese at Microsoft and who has good deadpan.
• "Looking Sideways", about many aspects of making things. A favorite episode: an interview with Deb Chachra.
• "StartUp", from Gimlet Media, telling the behind-the-scenes stories of people starting businesses. A favorite episode: "Shadowed Qualities" (although I think it won't hit you as hard if you haven't listened to previous episodes about Gimlet Media and founder/host Alex Blumberg).
• "Election Profit Makers", in which friends (one of whom is David "Get Your War On" Rees) rant about the 2016 US election season and swap tips about how to bet profitably on the election. I found out about this via MetaFilter and my Dreamwidth acquaintances and have barely listened so far but enjoyed what I've heard.