What is a self-contained proof? For this question I am required to give a self-contained proof of a statement, but I am not sure what a "self-contained proof" is. 
 A: Without knowing more context, it is hard to tell what was meant.
A proof is "self-contained" if it doesn't make use of external results.
The definition of "external results" is the variable in the question. For example, it could mean any of the following:

  
*
  
*Using only axioms
  
*Not using other results from this chapter
  
*Not using advanced results
  

I'm sure there are other possible meanings.
The first is really the only "formal" definition of "self-contained proof," but it is hardly ever what is intended unless you are studying foundations.
One example would be:

Prove that $1+3+5+\dots +(2n-1) = n^2$.

You could prove this using the fact that $1+2+3+\dots + n = \frac{n(n+1)}{2}$. That would not be self-contained, because you reference another result.
Alternatively, you could prove the theorem directly by induction, which would be a self-contained proof.
Quite often, during a chapter, we prove a very general result, and often an exercise will ask you to show such a specific instance of that result without resorting to the general theorem, just so you can see how a specific instance works out.
A: A self contained proof in its most rigorous form would be a proof that does not rely on theorems or lemmas that have not been proved previously in the same text. 
But in practice if you are asked to give a self contained proof I would expect that as long as you don't prove it by using generalizations of the theorem in question  or relying on complex theorems that most people reading it wont understand it should be considered a self-contained proof. 
The point of the self-contained proof is to reduce the prevous knowledge requirements as much as possible.
