What is the correct answer choice ? And what do I need to understand in order to solve this problem ?

You are asked for the relationship between A and P, so you must use the given information to write an equation for P that does not involve B. In order to do that, you need to find B in terms of A. This means you need to be able to write an equation that means
.. "the sum of two numbers A and B is 20"
and you need to be able to solve it for B.

You also need to be able to write an equation that means
.. "let P represent the product of A and B"
and you need to be able to substitute your expression for B into this equation.

.. A + B = 20 . . . . . the sum of A and B is 20
.. B = 20 -A . . . . . . solved for B by subtracting A

.. P = AB . . . . . . . . P is the product of A and B
.. P = A*(20 -A) . . .. substitute the above expression for B
.. P = -A^2 +20A . . the equation of a parabola that opens downward

Once you have the expression for P in terms of A, you need to be able to recognize (a) that it represents the equation of a parabola; and (b) that it opens downward (is concave downward).

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It is also helpful if you understand that for A and B to be non-negative, their values are restricted to the interval [0, 20].

You have to be able to overlook the fact that the wording "the relationship between A and P" is non-specific* as to which of those variables is the independent variable. Based on the work you have done so far, you are to assume that A is the independent variable, so the curve indeed opens downward, and not to the left.

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* This lack of specificity seems to be characteristic of recent texts built on the Common Core model. It is unfortunate that you are left to guess what the problem is really asking.

A + B = 20
Then, B = 20 - A


P = AB
P = A(20 - A)
P = -A² + 20A

It is the form of equation y = ax² + bx + c where a < 0
Then, the curve is parabola opening downward.
Since A is non-negative, the parabola is concave down.

The answer: D. Parabola concave down

A+B = 20
A = 20-B
P= AB
P = (20-B)B
P = -B^2 + 20B
compare with

y = -x^2 + 20x ----> where y= P and x= B
Parabola with negative slope
This is a parabola and is concave down as can be seen in the graph
http://www.wolframalpha.com/input/?i=graph+-x%5E2%2B20x

or
dy/dx = -2A
d^2y/dx^2 = -2 < 0 so it is concave down

P = A(20-A) = 20A - A^2;
it's a parabola concave downward.
That's not just the "best" representation,
it's a completely accurate representation.