Jacobian is an equivalent of a gradient vector for a vector-valued function. In an oversimplified way, you can think of it in the same way as the derivative of a scalar function.
You don't need to specify the Jacobian matrix for ODE functions in MATLAB explicitly. However, providing it will increase the speed because otherwise, the MATLAB will have to estimate the Jacobian matrix using finite difference, which essentially involves nudging each variable a little bit and estimating its effect on the output vector. If you explicitly provide it, then MATLAB does not have to go through all that nudging.
