Entertainment - Movies
How Did This Get Made - reviews of mostly older, bad films. They sneak a few new ones in (native advertising?) One of the funniest shows made. I've re-listened to many episodes multiple times. The humor might not be for you, but I'd recommend The Wicker Man episode as an excellent representation of the show, even if it isn't my favorite.
Film Sack - Similar to the above, review old/bad movies and make fun of them
F This Movie - movie reviews of older films with discussion about newer movies thrown in
Film Junk - focus on newer releases and movie news
Current Events
No Agenda Show - "The best podcast in the universe", bi-weekly breakdown of current events, kind of gone downhill the last few months, but still good and I have hopes for it's recovery
Common Sense with Dan Carlin - irregularly produced show about an issue in the news with Dan Carlin's opinion
Fresh Air with Terry Gross (NPR) - daily interviews of things in the news or interesting topics
KERA's Think podcast - similar to Fresh Air but produced by an NPR station, not as prominent as Fresh Air
Wait, Wait, Don't Tell Me (NPR) - Weekly news quiz show
On the Media (NPR) - media looks at the media
General Interest
Planet Money (NPR) - stories with some aspect relating to the economy, but more like 'pop economy'
Freakonomics Radio - Show by two economists looking at issues from their viewpoint, which is a horrible description for a show you have to try because you will probably love it
99% Invisible - excellent show that focuses on design and human interaction with it, to get an idea of it, I'd actually recommend watching the host's TED talk about flags
Still Untitled: The Adam Savage Project - Podcast with Adam Savage talking about various topics, making things, trips, behind the scenes Mythbusters stuff
Car Talk - Not produced newly for years, this has best of clips from their two decades on air. light hearted humor and 'dad jokes' abound, feels pretty dated at times, but I like it
Mystery Show - made by Starlee Kine, long time contributor to TAL, most of the episodes are excellent so far, first season is done, no word when the next one will start
Reply All - made by two guys that formerly made the podcast 'TL;DR' for On The Media. They took their show to the Gimlet Network and renamed it. The last few episodes are not their greatest, I recommend starting from the beginning, not working your way back.
Making It with Jimmy DiResta - Jimmy DiResta and friends discuss making things, not for everyone, but if you like DiResta's videos on YT, then it's an interesting dimension to hear these people discuss their creative process
Joe Rogan Experience - in depth interviews with people Joe Rogan finds interesting
History
The History of Rome - In depth history of Rome, as the title implies
Revolutions - The same host as the above, focusing on Revolutions in history. Just finished up the French revolution, next is the Haitian revolution
The Memory Palace - short stories from history, 10-20 minutes long, stuff you've probably never heard of
Stuff You Missed In History Class - Not very in depth, but a survey of different, often obscure, stories from history
12 Byzantine Rulers - profiles of 12 Byzantine Rulers... it's much better than it sounds. This is a very old school podcast
Norman Centuries - made by the same person as above, but focus on Norman history
The Ancient World - gonna start listening to this in the near future, but I have high hopes for it. It's a series about ancient Sumerian rulers
Dan Carlin's Hardcore History - excellent, in depth looks at events in history. Not as much a podcast as a really excellent history lecture. Dan Carlin could have toured with some of these stories and had people come see him perform them on stage, just sitting their speaking... I'm not kidding. Wrath of the Khans is a great place to start of the episodes available for free.
1,001 Heroes, Legends, Histories & Mysteries - Kind of a 'one man' Stuff You Missed In History Class, link is to the Stitcher listing for the podcast
The History of the Crusades - currently making my way through this. I find the subject matter interesting, but the audio quality is really rough so far, the information is good, though
Stitcher is great for listening to podcasts that you aren't interested in downloading. It's a great way to try out podcasts without subscribing. There are so many more that I listen to intermittently or that I used to listen to and stopped, but are still things other people really like. like Kevin Smith's Babble-on, a weekly look at the latest in pop culture news. It got too formulaic for me, but it's a good show for what it is. There are so many comedians with podcasts it could be considered a genre of podcasting.
Keep looking, you will find stuff you really enjoy. It doesn't have to be the mainstream 'podcasts', time shifted NPR shows professionally produced or repackaged radio shows. There are some really great and obscure shows out there, and more all the time as it becomes easier for people to make their own shows. One of the greatest things about podcasts is that it lets anyone have a show that is a couple of clicks away from being heard by anyone else in the world. Some would argue that is one of the worst things.