Note: for listening on iOS, you can use the apple podcast app which is fine, but I recommend getting either Downcast or Overcast which are both great podcast listening / management apps.
http://www.stuffyoushouldknow.com/
Probably the first one i started with. A must.
Stuff You Should Know, often abbreviated as SYSK, is a free, award-winning podcast and video series published by the HowStuffWorks website and hosted by Josh Clark and Charles W. "Chuck" Bryant, both writers at HowStuffWorks. The podcast is consistently ranked in the Top 10 on iTunes. It is one of the most popular podcasts in the world, being downloaded millions of times each month.
The podcast, released every Tuesday and Thursday, educates listeners on a wide variety of topics, often using popular culture as a reference giving the podcast comedic value.
http://www.radiolab.org/series/podcasts/
Also a classic and a must listen
Hosted by Jad Abumrad and Robert Krulwich, the show focuses on topics of a scientific and philosophical nature. The show attempts to approach broad, difficult topics such as "time" and "morality" in an accessible and light-hearted manner and with a distinctive audio production style.
Radiolab received a 2007 National Academies Communication Award "for their imaginative use of radio to make science accessible to broad audiences".
http://99percentinvisible.org/
Amazing one about architecture and design.
The show's name is taken from a quote by Buckminster Fuller: "Ninety-nine percent of who you are is invisible and untouchable." Its goal is to expose the unseen and overlooked aspects of design, architecture, and activity in the world.
Each episode generally focuses on a single topic or specific example of design, often including interviews with architects, experts, or people who have been influenced by the design.
https://gimletmedia.com/show/surprisingly-awesome/
One of my newer favs.
Surprisingly Awesome is an American podcast formerly hosted by Adam Davidson and Adam McKay and currently hosted by Rachel Ward. In each episode, one of the hosts takes a seemingly boring topic, and try to make the other believe that it is interesting.
http://www.thisamericanlife.org/
Again, a classic. Originally was a radio show on NPR and now a super popular podcast.
This American Life (TAL) is an American weekly hour-long radio program produced by WBEZ and hosted by Ira Glass. It is broadcast on numerous public radio stations in the United States and internationally, and is also available as a free weekly podcast.
Primarily a journalistic non-fiction program, it has also featured essays, memoirs, field recordings, short fiction, and found footage.
http://freakonomics.com/archive/
Super interesting one on random economics / politics / social issues.
Freakonomics Radio is an American public radio program which discusses socioeconomic issues for a general audience. The show is a spin-off of the 2005 book Freakonomics. Journalist Stephen Dubner hosts the show, with economist Steven Levitt as a regular guest.
http://gimletmedia.com/shows/reply-all
All about the internet!
Reply All is an American podcast from Gimlet Media, hosted by PJ Vogt and Alex Goldman. It features stories about how people shape the internet, and how the internet shapes people.
http://www.npr.org/planetmoney
Nice short one that takes complicated economic issues and makes them understandable.
The podcast launched on September 6, 2008 to cover the global financial crisis of 2008–2009 in the wake of the Federal takeover of Fannie Mae and Freddie Mac. It was created after the success of "The Giant Pool of Money", an episode of This American Life. Currently, episodes are produced two days a week and are around 15 minutes in length.