a trajectory

If the phase plane is a map of all possible states, a trajectory is the actual route one particular run of the system takes across that map. Start the system from some chosen state — a definite (x, y) at time zero — let it evolve, and the state point traces a single curve. That curve is the trajectory, also called the orbit. It is the answer to 'starting from here, where does the system go?', drawn as a path rather than written as a formula.

Precisely, a trajectory is the image in the phase plane of one solution (x(t), y(t)) of the system, with time stripped away — you keep the shape of the path but forget how fast it was traversed. Crucially, in an autonomous system (one whose rules do not depend on the clock) two distinct trajectories can never cross: at any point the velocity field points in exactly one direction, so the state has no choice about where to go next. Trajectories may approach an equilibrium, spiral toward it, fly off to infinity, or close up into a loop, but they tile the plane without ever intersecting.

Trajectories are the raw material of qualitative theory. You rarely need the formula for (x(t), y(t)); you need to know which trajectory you are on and where it leads. Because trajectories cannot cross, sketching just a few of them — together with the equilibria they avoid or approach — often pins down the behaviour of every other one, which is the whole economy of the phase-plane method.