Step 1: Look at the equation \( \left(\dfrac{d^2y}{dx^2}\right)^3 + \left(\dfrac{dy}{dx}\right)^2 + y = 0 \) and pick out the highest derivative present, here it is \(\dfrac{d^2y}{dx^2}\), a second-order derivative, so the order of the differential equation is 2.
Step 2: Confirm the equation is a polynomial in its derivatives, with no derivatives trapped inside functions like sine or under a root, so that "degree" is well defined here.
Step 3: Read off the power to which the highest-order derivative, \(\dfrac{d^2y}{dx^2}\), is raised in the equation, it appears as a cube, so the degree is 3.
\[ (\text{Order},\ \text{Degree}) = \boxed{(2,\ 3)} \]