Interesting. So do you officially own the house or are your payments more like rent, and the Council actually owns it?
Sequoias: Here is how equity works. Say you buy a house for $100,000, with a low down payment of $3,000. You now have $3,000 equity (value) in the home, and a $97,000 mortgage. Let's say property values go up (as they used to do), and the next year, houses similar to yours sell for $110,000. It is fair to assume your house is worth the same as these others.
You now have an additional $10,000 in equity, plus the $3,000 you put in, plus about $1500 in principal that you would pay off in a year's time (out of your mortgage payments of principal plus interest). So you now have about $14,400 in equity, and have only paid for about $4500 of that; the rest came from the rise in value of your property.
As the years go on, you can see that if property values continue to rise, you will have increasing equity that is more than what you yourself have paid, although you will also be paying interest to the bank all this time, as well.
Conversely, let's start with the same example, but let's say property values go down by $10,000 in a year. Now you have a house worth $90,000, and a mortgage that is about $95,500 ($97,000 minus the $1500 you have paid in principal). You owe more than what the house could sell for, so you are "under water."
Banks really don't like that situation, so they would prefer that you come up with 10 or 20% down, so that you are taking somewhat more of the risk, as compared to them taking nearly all of it.
Being under-water on your loan doesn't mean a thing if you don't need to sell, though. It's all a hypothetical situation until you want to sell it. Most often, if you hold on to a property long enough, you will make money on it. If you have to sell before you have owned it very long, though, you might very easily lose money on it, especially considering closing costs, etc., that you will never get back.
After we pay off in the future, then we will own the house free from worry. It is almost like rent to own. It is the same like "All In The Family" where Archie and Edith burn the paper after they have finished the payment on the house and owned the house forever. No worry. Our payment for the house will be a long, long way to go. 
