Way back when I was just filling in as a guest host I was using a gaming headset to record for British Invaders. Once my guest spot became a regular deal I upgraded to a nice Blue Yeti usb microphone which has been OK so far. I use the free software Audacity to record my end of our discussions and then upload a wav file to Brian who stitches the two halves together and does all the editing magic.
It's a pretty good system that has worked well for us so far, but for Mega City Book Club I have a revolving panel of guests to record and obviously I am dependent on whatever kit they have or can lay their hands on. I'm using Call Burner for Skype which records both ends of our conversations as two separate wav files. So far I have carried on using audacity to edit the first four episodes but I am now learning how to use Sound Forge 10 as I strive for a more professional result.
I have also used audacity to turn my edited audio into the final mp3 files. I was advised by a couple of people that iTunes does a better job of this but strangely my results seem to have a lower quality than I wanted so I'm sticking with audacity until I master Sound Forge.
There's nothing like starting a podcast to send you to amazon to upgrade your kit and a recent spending spree has produced a Shure SV100 and a Focusrite Scarlett interface to connect it to my computer. My office is starting to look like a recording studio. I've also bought a Tascam digital recorder to use when I'm out and about. I used it to record episode four with Tony Richards which we did upstairs in a pub in central London.