Terminal Side
The terminal side of an angle in standard position is the rotating ray that ends at the angle's final position after rotation from the initial side.
Definition
The terminal side is the moving ray of an angle, the one you rotate to from the initial side (positive $x$-axis) to create the angle. Its position determines the values of all trig functions: the terminal side of $\theta$ intersects the unit circle at $(\cos(\theta), \sin(\theta))$, the ray from the origin through that point. As $\theta$ varies, the terminal side parametrizes points on $S^1$, and coterminal angles sharing a terminal side reflect the $2\pi$-periodicity of the map $\theta \to e^{i\theta}$ from $\mathbb{R}$ to $S^1$.
Example
For a $90^\circ$ angle, the terminal side points straight up along the positive $y$-axis; for $180^\circ$, it points left along the negative $x$-axis. The terminal side of $225^\circ$ lies in quadrant III, intersecting the unit circle at $(-\sqrt{2}/2, -\sqrt{2}/2)$, confirming $\cos(225^\circ) = \sin(225^\circ) = -\sqrt{2}/2$; the terminal side of $7\pi/4$ passes through $(\sqrt{2}/2, -\sqrt{2}/2)$ in quadrant IV, and this is coterminal with $-\pi/4$ and $15\pi/4$.
Key Insight
"Terminal" means ending, where the rotation stops, and two different angles can share the same terminal side (coterminal angles), which is why $\sin(\theta) = \sin(\theta + 2\pi)$ for any $\theta$. The terminal side is the geometric realization of the equivalence class of angles modulo $2\pi$: two angles are equivalent if their difference is a multiple of $2\pi$, and the terminal side represents the unique element of each class on the unit circle.