Look back at the damped oscillator x" + 4x' + 5x = 0. The characteristic polynomial, lambda^2 + 4 lambda + 5, has roots -2+i and -2-i, so the general solution is e^{-2t}(a cos(t) + b sin(t)), or, in polar form, A e^{-2t} cos(t-phi), where a and b are the legs of a right triangle with hypotenuse A and angle phi opposite b. You get a damped oscillation.
Create a system by declaring x' = y, y' = -5x-4y. The matrix has the same characteristic polynomial, whose roots have negative real part and nonzero imaginary part, so you expect a stable spiral. It spirals clockwise, because at (x,y)=(1,0) the vector field has the value (0,-5), pointing south. I drew it. Notice the alternation of signs of x and its derivative. These spirals are the solution curves of a first order ODE in x and y: by the chain rule,
dy/dx = y'/x' = -(5x+4y)/y.
The trajectories are solution curves for this ODE. The vector field is vertical when x'=0, i.e. when y=0; and horizontal when y'=0, i.e. when y=-5x/4.
Here is a summary of the nondegenerate linear phase portraits: If the eigenvalues are:
Real and distinct
Real repeated
Complex
On to nonlinear systems. The Volterra preditor/prey model: We have bugs and birds, with populations proportional to x and y. In isolation x' = ax, y' = -cy (the birds need bugs to thrive). Together, the birds eat the bugs; this decreases bug fertility, x'=x(a-by), and increases bird fertility, y' = y(-c+dx). The is a nonlinear system. Analyze it as we did with the spiral. The vector field is vertical where x'=0: x=0 or y=1. It's horizontal where y'=0: x=1 or y=0. This means that it's simply 0 when x=y=0 and when x=y=1. These are the "critical points" of the system. Along y=1 the vector field points north to the right of (1,1) and south to the left. Along x=1 the vector field points west above (1,1) and east below it. This indicates a spiral or center, centered at (1,1). I sketched in the phase portrait.
By the chain rule, dy/dx=(x-1)y/x(1-y). This is separable, and the solution is (x-ln|x|) + (y-ln|y|) = c. You don't want to solve this for y as a function of x, but you can use it to see that you do get closed orbits, rather than a spiral.
Now let's introduce an insecticide. It decreases the fertility of both bugs and birds: say x'=x(1-y-k), y'=y(-1+x-l). Now there are critical points at (0,0) and at (1-k,1+l): we have increased the number of bugs and decreased the number of birds, not a desirable or expected outcome.