This is what I currently subscribe to. I use Mobilecast to download them directly to my iPhone.

In no particular order:
This American Life - Really, if you haven't heard of this one by now, you probably haven't listened to radio in years. This show gets a lot of credit for pulling public radio away from its old image of curmudgeonly men with beards and brown blazers and aging hippies.
Radiolab - one of the many shows that's followed in TAL's footsteps, and arguably the best. Follows a very similar format, but usually with more science and philosophical discussion thrown in on top of the classic storytelling. They also seem to have a bigger budget for sound editing, so you don't just hear the same music over and over and over.
WGBH Classical Performances - needs no explanation, really. A bit more traditional classical than I'd like. I'm still searching for something more contemporary.
Sparkletack - The San Francisco History Podcast - this one's probably my favorite. Even if you're not familiar with San Francisco, it's very well done. Everything's researched, written, recorded, and edited by one guy in his spare time, so updates tend to be sporadic, but it's an excellent example of what a single person can do at very little cost in this day and age.
Bikescape - another one-man show, this time by a good friend of mine. All about bikes and bike culture, with a local slant.
Selected Shorts - how can you not love this one? Short stories and essays read by actors. Each show follows a theme: the last show I listened to, "Brave New World," included an essay by a Bornean herdsman visiting New York followed by a Phillip K. Dick story.
Live Concerts from All Songs Considered - a fairly wide variety of music here. A quick scan of the list reveals shows by Bjork, Yo La Tango, and REM, among others.
Love and Radio - another TAL copycat, this one's often fairly short and usually far more twisted. It comes from "alt.NPR," which appears to be what was left on the cutting room floor for being too risque, too disturbing, or just too weird to make it to the radio.
In Character - brief essays about archetypal characters in fiction. Recent shows have been on Hester Prynne, Indiana Jones, and Eric Cartman.
Poetry off the Shelf - essays on poetry and movements within poetry.
Garrison Keillor's The Writer's Almanac - don't let the host turn you off: this has nothing to do with A Prairie Home Companion. Keillor gives a daily snippet of literary history and reads at least one poem. I find it an excellent way to start the day.
Living on Earth - environmental news that I keep trying to get around to listening.
The Changing World - world news that I keep trying to get around to listening. D'oh!

Mind you, I don't necessarily listen to all of these on a regular basis, but they're great to keep around. The only ones I follow religiously are TAL, Radiolab, and Sparkletack. I'd love to find some new ones, so if anyone has something to recommend, I'm wide open.