I understand that the Runge Kutta is simply an extension of the Euler Method, which is:

$\displaystyle y_{i+1} = y_i + f(x,y)h+\frac{1}{2!}f'(x,y)h^2$

But then I don't see how to get $\displaystyle k_1$ and $\displaystyle k_2$? Also in some text books, what does this partial derivative mean?

$\displaystyle \frac{\partial{y}}{\partial{x}}|_{x_i, y_i}$

What is up with the vertical bar and the variables at the bottom?