How to classify equilibrium points
Classification of equilibrium points is done on the basis of the eigenvalues.


If the two eigenvalues have no real parts, it is a hyperbolic fixed point and represents undamped oscillation.
If both have a negative real part, it is a stable fixed point. If any of the eigenvalues has an imaginary part then it represents damped oscillations (in that case the equilibrium point is called a focus).
If one or both eigenvalue have a positive real part, it is an unstable fixed point.
If one eigenvalue has a positive real part and the other a negative real part, it is a saddle point.*


You sketch the trajectories of the equilibria by starting a tiny distance away from the point (where it is not in equilibrium) and trace the quiver plot. Drawing trajectories around unstable points is tricky and you have to plot the function in negative time i.e. instead of plotting f(x1,x2,t), you plot -f(x1,x2,t).  This page has representative trajectories for the different types of equilibrium points.  This page has a more thorough explanation, generalized to multiple dimensions.

*This classification appears to be slightly variable (more or less categories), on the basis of the two links I found.  I think the one I list is the simplest version, but any classification should suffice.  They are all based on the same principles. You can refer to some books on non-linear dynamics:


Nonlinear Systems by Hassan Khalil
Nonlinear Dynamics and Chaos by Steven Strogatz