What is a localiser back-course approach and how does it work?
A localizer back course is simply flying on the back side of the localizer to the runway.  How a localizer works is it sends out two signal lobes:  one at 90Hz and the other at 150Hz.  The localizer needle in the airplane compares the relative strength it receives from both lobes and then shows a deflection.  

See this figure from the Instrument Flying Handbook (2015) published by the FAA.


Notice the top left of the picture shows the same lobes on the same side of the runway.  Now imagine flying the front-course, the blue lobe is on the right.  If you are flying the back-course, the blue lobe would be on the left.

The localizer needle doesn't know if you are on the front-course or the back-course and will show the same deflection for both situations.  The problem comes in for the back-course is that the needle deflection is deflected the wrong way.



To answer your questions.


What does back-course mean?  It simply means you are flying on the opposite of the localizer signal.  If there is a back-course published, the signal strength will be strong enough to fly the approach as required. 
How common is a back-course approach?  There are currently 69 back-course approaches in the United States.  There are approximately 1,893 ILS approaches.  This is around 3.5% of back-course approaches as compared to ILS approaches.
The same flight and navigation instruments required to fly a front-course ILS is required to fly a back-course ILS.




From a technique standpoint.  Whenever I fly a back-course approach, irrespective of whether I am using an HSI or VOR receiver, I will always set in the front course.  For pilots that use an HSI this will provide for "positive sensing" meaning the needle will deflect correctly.  For pilots using a VOR, the needle will shade bottom numbers that correspond to heading that need to be flown to get back onto the back-course approach.



If this was a back-course approach, notice that the needle is deflected to the right.  Conventional wisdom says you need to turn away from the needle to re-intercept.  I say to select a heading under the needle to fly.  That heading will be approximately 120°.  Take a look at the heading indicator and determine that you most likely need to turn left to re-intercept the back-course.  Try it and you will see it works.  Conversely, I also teach the same concept for an ILS except the heading to fly is on top of the VOR indicator instead of on the bottom.