Runge-Kutta_fehlberg 4th/5th is an adaptive stepsize algorithm. It adjusts the stepsize in such a way as to keep the absolute error in a step less than Absolute Tol. and the fractional error less than Relative Tol. See Numerical Recipies in C: The Art of Scientific Computing, by Press, Teutolsky, Vetterling, and Flannery, for more information on the algorithm.
The Rosenbrock method is an adaptive, semi-implicit, multi-stage method . The Rosenbrock method can be thought of as a generalization of the Runge-Kutta methods. This is done by adding more parameters into the equations for this method. As a result of this generalization, the Rosenbrock Solver can solve stiff differential equations. This is something that the Euler Method and the Runge-Kutta methods cannot do well. Like the RKF45 method described above, it adjusts the stepsize in such a way as to keep the absolute error in a step less than Absolute Tol. and the fractional error less than Relative Tol.
See the description of the RKF45 options above.
The Euler algorithm is a very basic first-order, fixed step size ODE solver. It is included in ODEToolkit merely for demonstration purposes as it is inaccurate.