Solving second-order ODEs with Taylor series approximation of second derivative and linear algebra
What you have there is the central difference approximation to the second derivative. The Wikipedia article on finite differences explains this pretty well.

Of course, this isn't a method as such; that depends on the exact problem being solved. If this were an initial value problem, then a simple rearrangement of this equation is typically known as Störmer–Verlet integration (or just Verlet integration depending on whom you ask).

If it was a bounding-value problem, then this discretisation of the derivative gives you a tridiagonal system, and you don't have a complete "method" until you have a method to solve it. A small modification of Gaussian elimination is quite efficient on matrices of this form.

As to competitors... that's a very long discussion. Any introductory textbook on numerical analysis should give you a good start.