I'm very, very behind on my podcasts: I try not to skip too much but I seem to download more than I have time to listen to so I'm about a year behind, right now. So this list might include things that aren't as good as they used to be, I guess...
Not all of them are safe for work, mostly on language grounds. Google them by all means but I won't post links, for that reason.
Answer Me This
These two usually make me laugh. The latter would be topical if I were up to date but it's worth it for John Oliver -- yes, that John Oliver -- and Andy Zaltzman's unique take on world politics. The former is ... well, the clue is in the name. It's also usually funny.
I'm not sure of availability outside the UK but the BBC produces quite a few weekly podcasts that I like. In particular, the Friday Night Comedy podcast, which is always news-based comedy, and More Or Less, which is about statistics, often the sort that are misquoted by politicians and journalists, and is usually a lot more interesting than you might think. (There's a version of it that's broadcast on the World Service and I imagine the podcast of that one is available everywhere, even if the others aren't.)
And I've been recommended Radiolab, but apart from hearing a bit of one in a documentary podcasts about podcasts (!) I have yet to hear a full one. I have a few downloaded, though...
I've recently reluctantly given up on Escape Pod. Excellent, good quality sf stories but I have to try to clear the backlog somehow... and Greg Proops' Proopcasts were often amusing (also somewhat NSFW) but patchy and his lengthy diatribes on the subject of baseball aren't a good fit for British ears, so I, possibly also unfairly, gave up on those too.
Mostly, podcasts are for car journeys and distracting me from the tedium of things like vacuuming; I have a rolling playlist that ensures I listen in chronological order and -- with extremely rare exceptions -- they're deleted after listening.