Okay, okay, I see that you genuinely didn't know foureleven
Swift hijack, then conversation can return to the OP's dilemma 
Council housing is, as the name implies, any housing owned and kept by the Local Authority. Gnereally, there is a misconception that you have to be very poor or unemployed to be allocated a council house. This isn't true. Anyone can join the 'list', but some LAs operate a 'points' system, and if you are not in a vulnerable group your points are unlikely to reach the levels needed for a council house. Points are allocated to a tarrif, and the cumulative total determines your place on the list.
Other LAs use a 'bidding' system, whereby potential tenants are allocated a band from 1-4, where 1 is 'emergency' i.e. top and 4 is 'no priority', band 4 would be a family who are currently adequately housed and can afford their rent, for example. A property comes onto the bidding list, and people bid for it. Band 1 trumps 2, band 2 trumps 3, band 3 trumps 4. If there are more than one band 1 bidders, the house is allocated on length of time on the list. In theory, it is fairer, because only people who want to live in that location, etc., will bid. In theory, a band 4 person could get lucky, if no band 1,2 or 3 people bid. The reality is that housing stock is in such short supply that every house is bid for by hundreds of people.
Housing benefit is payable to anyone with an income below the 'applicable amount' for their particular family make-up and circumstances. Each applicant is awarded an amount that they are expected to live on. This will be made up of a series of elements. I.e. Lone Parent or Couple (same amount either way) + x children + Disregard of earnings + Disability premium ... and so on.
Then all of the applicant's income is totalled. They compare the income to the applicable amount, calculate 65% of the difference, and that figure is their 'excess'.
The LAs set a local housing allowance for each Broad Rental Market Area, which is the cap for rental of that particular size of house.
The Housing benefit you get is the LHA-the 'excess'.
Council houses are generally cheaper than private rentals, because councils don't make a profit on them. They charge capital value plus maintenance costs. So, generally, if you are on a low wage, you will get very much less HB than if you were in private rental, because you would have a much lower rent.
So, although the likelihood is that a fair proportion of families in council housing may be without a waged income, it is no less so than the private sector.