One of my favourite podcast genres is actual play Dungeons and Dragons. They're chock full of hilarity and surprisingly deep emotions. It's hard to pick just one to recommend, some are pretty light and accessible while others are so long and dense that they're a huge time commitment.

Adventure Zone

The McElroy brothers from My Brother, My Brother and Me are joined by their father, Clint, in DnD sessions run by Griffin. Very fun, but starts a little silly and then progresses to an amazing story split into cool arcs.

Not Another DnD Podcast

Also fun and chaotic, this podcast introduces you to Brian Murphy as the DM and Emily Axford, both of whom appear in Dimension20 below as players. The PCs in this are great characters, my favourite being Beverly.

Dungeons and Daddies

I think if I had to recommend a single podcast I would be tied between Dungeons and Daddies and Dimension20 Fantasy High at the moment. Just like all the rest of the podcasts here it is both laugh-out-loud funny and also surprisingly sweet and emotional. The theme of 4 real-world dads thrown into a DnD setting works really well.

Maybe better to watch

Yeah, I said podcasts, but these are great actual play YouTube shows that are probably better to watch.

Dimension20

While most of Dimension20 is available on DropoutTv for a small subscription, the entire Fantasy High season is up for free and it's awesome. Brennan Lee Mulligan runs a game for a party of six. Lots of fun but it's impressive watching how fast Brennan and the players can create real family bonds with NPCs.

Critical Role

Disclaimer: I haven't watched this.

Matt Mercer is the DM for these huge sessions. Fans absolutely love Critical Role and the seasons are all up on YouTube as far as I can tell.