I have heard of the  Music and Comedy Festival for many years, mostly from Craig Ferguson when he had his Late Show here in the US. The little bit of research is showing me there are many festivals, and the one I think I hear about takes place most of August.  There seem to be a parallel Fringe Festival as well.

Can anyone share info?

@Cargo

Just seen this!

The Fringe is the one you seek - although the MAIN Original Edinburgh Festival also encompasses the Edinburgh Military Tattoo - which I would be taken to most years as a kiddie.
Worth seeing at least once if you can manage it. Not expensive either.

Lots to see and do in Edinburgh during August every year - BUT - if you don't plan the accommodation about a year ahead, you'll be paying a fortune to stay in normal cheap hotel type rooms
(I've seen daily rates posted for some normal Ibis/Travelodge/Holiday Inn places, of over £450 per night - some were a lot more)

It may be cheaper to look outside the Edinburgh city area slightly - places such as East Lothian are still within 30 minutes - or even places in the Borders such as Galashiels or Melrose = but again, these areas fill up quick as August is UK family time too, so normal demand is there PLUS the Festival attendees too. These areas are well served by good cheap public transport networks. East Lothian has some good inland areas, lots of history plus coastal golf courses that host The Open. And a lowland distillery at Glenkinchie - the tour there is well worth the small fee. Samples, lots of samples!

I have just moved away from East Lothian back to England - but if you get stuck I still have a few friends in the area who can ask around for some off the beaten track type places to stay.

Get in touch with the Fringe office & they will send you the full itinerary for the month, plus the British Tourist Information folk can not only advise on where to stay, but a lot of the time they will arrange the booking for you.

PM me or post in here if you need more info

ETA: Got Craigs autograph about 30 years ago. He was doing support to Harry Enfield, just starting out. A genuinely nice bloke to chat to. Got a surprise when he hit the big time a couple of years later - huge star now.