Recording Skype conversations is fairly straightforward

You will need a third party software to do this. All promise to record, but some only record your voice, not the incoming - so check carefully.

I think the simplest software I have found to date is

This promises to do all that I want, and easily and free.
It integrates into Skype by adding a menu function allowing recording to be initiated automatically or on request.
It records both parties on different tracks so it's great for podcasters who are trying to get just one side of an interview
Which Audio formats does PowerGramo support?
Now PowerGramo supports Ogg Vorbis and wav. By default, PowerGramo uses Ogg encoding format, which we think is very efficient and produces high quality audio.

How can I edit the recording?
You can export your recordings and edit them in any other Audio editing tools. We recommend Audacity, which is open source and free.