The great thing about podcasts is that there is something for almost anyone. Here are some of the podcasts I listen to, sorted by how quickly I devour the episodes:

- 99% Invisible: A+ podcast about all kinds of designs, from flags to buildings to submarines. Almost never fails to pique my interest.

- Radiolab: Great stories, amazing audio design. Headphones recommended! I was introduced to Radiolab with the Speed episode. Perhaps the memorable episode I ever listened to is Bliss.

- Song Exploder: Pulling apart songs with the composer's commentary. If you know the songs, they're great episodes. For example, Game of Thrones or House of Cards themes.

- This American Life: The elephant in the room of podcasting. Has some amazing episodes. Lately they've been doing a lot of reruns and some 'meh' episodes, but they have a huge archive of great episodes, like Death and Taxes

- Invisibila: only one season but off to a great start. I found How To Become Batman very intriguing.

- Dan Carlin: Hardcore History episodes are basically free, small audiobooks about an aspect of a period of history. I know that doesn't sound appealing, but Carlin is good enough to make anything interesting. His Common Sense podcast discusse about current event from a geopolitical / historical point of view.

- The Bugle: John Oliver and Andy Zaltzmann discuss the week's news with humour (much funnier than Last Week Tonight, if you ask me). They're currently on a break, but you can listen to the last regular episode, 291, here.

- The Gist with Mike Pesca: episode every weekday, where Pesca opines on / discusses important news and good articles, often with authors or experts. I don't listen to all of what he has to say but he almost always has interesting points (if he's not being silly). Latest episode is quite representative.

I also listen to Cortex and Hello Internet, but almost entirely because I like the hosts. Start from the first episodes, it doesn't get much different than that to be honest.

edit: I forgot to directly address your question, whoops. My ideal podcast is either thought-provoking, emotive or funny, and preferably a combination of the three.