For a download file reviewing in some detail the main types of trajectories, see the link below.
Here is a brief overview, including some pictures.
Saddle Point.
This is the case in which the eigenvalues are real and have opposite sign. In this case the trajectories are asymptotic to the positive eigenspace as t approaches infinity, and to the negative eigenspace as $t$ approaches negative infinity.
Node
This is the case in which the eigenvalues are nonzero, real and distinct and have the same sign. If the sign is positive, the trajectories move away from the origin with increasing time, and the node is said to be ``unstable.'' If the sign is negative, the the trajectories approach the origin as $t$ approaches infinity, and the node is said to be ``stable.'' In either case, the trajectories are tangent to the eigenspace corresponding to the eigenvalue with smaller absolute value as they approach the origin.
Degenerate Node
There are various special cases called ``degenerate nodes.'' For example, if the eigenvalues are equal and not zero and the matrix is diagonal, the trajectories are rays emanating from the origin. If one of the eigenvalues is zero, the trajectories are rays parallel to the nonzero eigenspace, but also each point on the zero eigenspace is a trajectory corresponding to a constant equilibrium solution. If both eigenvalues are zero and the matrix is diagonalizable, it is the zero matrix, and every trajectory is constant.
One negative and one zero eigenvalue
Spiral Point
This is the case arising from complex eigenvalues. If the real part is positive, the trajectories sprial out with increasing time, and if they are negative, they spiral out with time. If the eigenvalues are purely imaginary (real part = 0), then they are periodic, and form closed ellipses. You can detect whether the motion with time is clockwise or counterclockwise with time from the shape of the matrix A, by looking at the vector field. (See the examples below and also the .pdf file for more on this.)
Improper Node
This corresponds to nondiagonalizable matrices, and is an interesting transitional case. In this case there is a single eigenvalue and a one dimensional eigenspace, that is, a line L. If the eigenvalueis positive, the trajectories move away from the origin as time increases and are unbounded.The slope of the trajectory approaches the slope of the line L (but is not asymptotic to it, contrary to what it says in the book).As time approaches negative infinity, the trajectory approaches the origin and becomes tangent to the line L. Most of the trajectories make a semiloop, as you can see from the graph.