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

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 k_1 and k_2? Also in some text books, what does this partial derivative mean?

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

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