Correct me if I am wrong but this means that a firm producing a product keeps producing ever increasing quantities up to the point where the price has come down by so much that it doesn't cover the cost of production any more. Is this correct?
I think you are confusing two different concepts. First of all you have to differentiate between long-run and short-run. Are you talking about total fixed costs or total variable costs? If average variable costs are greater than the marginal costs, the firm will stop producing. These are not directly seen on a supply and demand curve you discribed.
Assuming it is, I would like to know how this looks like in real life because it is this part that my mind is having trouble with. Because for this concept to work, it would mean that the firm would be selling the same product at a whole range of different prices which of course makes no sense since who would want to buy something at a higher price if its also available at a lower one.
No, the supply and demand curve is a description of that particular product and the market for that product. At such and such a price, a firm would be induced to produce a certain quantity of product and consumer could possibly demand a different quantity. When the price is at equilibrium, quantity demanded = quantity supplied.
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