All Songs Considered: NPR's music podcast hosted by the sweet and lovely Bob Boilen and his depressive coffee addict co-host Robin Hilton. They play a wide range of music so its a really great way to learn about new music that you wouldn't necessarily hear about. Also check out their Tiny Desk concerts.
Song Exploder: A fantastic music podcast where "musicians take apart their songs, and piece by piece, tell the story of how they were made. Each episode is produced and edited by host and creator Hrishikesh Hirway in Los Angeles. Using the isolated, individual tracks from a recording, Hrishikesh asks artists to delve into the specific decisions that went into creating their work." Each of the episodes are totally unique and which ones you like will be based on your music tastes but a few of my favorite episodes are: Sylvan Esso, Spoon, DJ Shadow, Bojack Horseman, & the Reply All Exploder episode.