What does an individual's demand function describe? Now, this seems a super easy and straightforward question, right? From wikipedia: 


  The demand curve is the graph depicting the relationship between the price of a certain commodity and the amount of it that consumers are willing and able to purchase at that given price. 


To me, this reads as follows: for an individual, the demand function answers, for all values of $p$, the question: "if the price is $p$ per unit, how many units $q(p)$ would the individual buy?"

As an example, let's consider my personal demand curve $q = 10-p$ for a certain good. I would interpret that, also with the above definition, as follows: if the price is 10/unit or more, I purchase nothing. If it is 7/unit, I'll purchase 3 units. 

The big point here is that this seems to imply, that I'd pay 7/unit for each of the 3 units. Is that correct? Things would be different if I were faced with a situation in which I had to pay more for the first 2 units. In that case, I might decide to buy less than 3 units. And I wouldn't be contradicting my demand function. At least, not, if it were as I interpreted the definition.

Alternative: it seems less close to the text, yet much more convenient to the mathematics, to interpret the demand function in the following way. The demand function answers, for all values of $q$, the question: "if an individual already has $q-1$ units, how much would he be willing to pay for unit $q(p)$?" (or, as an integral, "if an individual already has an amount $q$, how much would $p(q)$ be, with $p(q)dq$ the amount he's willing to pay for an additional amount $dq$?").

Thanks!
The demand function does not say how much he's willing to pay for an additional amount dq. The demand function says: given the price of good is $p$, you will purchase $Q(p)$ of them. I think your confusion lies in the fact that it is possible to charge different prices for each unit of good. Notice that our theory is assuming one price, not multiple.

Moreover, the notion that consumer surplus measure the difference between what you paid and what you are willing to pay is completely false. Suppose the demand function is $Q(p)=-5p+100$. Suppose further that the price is $5$, so you would be purchasing at a quantity $75$. The consumer surplus is: $$CS=\int_0^5Q(p)dp=\int_0^5 -5p+100dp=562.5.$$ If interpreting it as the difference between what you paid and what you are willing to pay, we know that you are willing to pay $562.5+5\times75=937.5$ for $75$ units of them. In other words, you are willing to accept a price $937.5/75=12.5$. However, plugging this number into our demand function, we find that $Q(12.5)=37.5$, i.e. you are only willing to buy 37.5 units of the good, which clearly contradicts to our interpretation of consumer surplus.

For a better treatment of consumer surplus, I would recommend you to consult advanced texts on microeconomics theory such as Mas-Colell's or Varian's more advanced book.