The order of a numerical approximation method, how to calculate it, and comparisons What does the order propriety say about a numerical approximation method?
How can I know, having a runge-kutta butcher-tableau, which order does this method have? 
How to calculate it? 
How would you compare the order two runge-kutta-methods, when you're given two extended runge-kutta butcher-tableau's of each one? 
How is possible to compare the error-order of both?
 A: When speaking of differential equations, a numerical method aims to compute a function table that represents an approximation of the solution function. The table is computed for a discretization/subdivision of the domain of the solution. The cells of the subdivision have some characteristic length $h$.
The order tells you how big the error is with respect to the discretisation. With order $p$, a refinement of the subdivision by a factor of $2$ (in all directions) will reduce the error by a factor of $2^p$.
You can not see the order from the structure of the tableau. A tableau of $s$ stages can have orders from $0$ to $2s$.
To validate an order $p$, the order conditions up to that order have to be satisfied for the tableau. See Butcher trees on how to systematically compute these polynomial expressions. https://www.math.auckland.ac.nz/~butcher/ODE-book-2008/Tutorials/
Given two methods of the same order, further comparisons are more difficult to obtain. One can compare the residuals of the order conditions of order $p+1$. One can do benchmarks with differential equations of different characters. This gets even more experimental when adding strategies for step size selection.

Usually in the design of RK methods one tries to reduce the size of the order expressions for the next order by imposing additional conditions on the entries of the tableau. As "optimality" is a multi-valued objective, this optimization does not have an unique minimum, not one method (of a given order) that is best at everything.
