Displacement
Displacement is the straight-line change in position from start to finish, a vector with both magnitude and direction.
Definition
Displacement is how far an object ends up from where it started, and in which direction: the straight-line shortcut from start to finish rather than the winding path. It is the change in position, $\Delta x = x_{\text{final}} - x_{\text{initial}}$, a vector quantity with both magnitude and direction, and displacement divided by time gives average velocity. In vector form it is the difference $\Delta\vec{r} = \vec{r}_f - \vec{r}_i$ between final and initial position vectors. Displacement is independent of the path taken, which is why it obeys vector addition: successive displacements add head to tail. Its time derivative is velocity, and its integral form is $\Delta\vec{r} = \int \vec{v}\, dt$.
Example
You walk $40\ \text{m}$ to the water fountain and $40\ \text{m}$ back; your distance is $80\ \text{m}$, but your displacement is $0$ because you ended where you began. A hiker who walks $3\ \text{km}$ north and then $4\ \text{km}$ east has a displacement equal to the straight line from start to finish, $\sqrt{3^2 + 4^2} = 5\ \text{km}$ pointing northeast, while the distance was $7\ \text{km}$. A robot that moves $\langle 6, 0 \rangle\ \text{m}$, then $\langle -2, 5 \rangle\ \text{m}$, then $\langle 0, -1 \rangle\ \text{m}$ has a total displacement of $\langle 4, 4 \rangle\ \text{m}$, magnitude $4\sqrt{2} \approx 5.66\ \text{m}$ at $45^\circ$, while its total distance is $6 + \sqrt{29} + 1 \approx 12.4\ \text{m}$.
Key Insight
Displacement can be zero even after a long trip: distance tells the story of the journey, and displacement only cares about the ending compared to the start. The magnitude of displacement is never larger than the distance traveled, and the two are equal only when the object moves in a straight line without turning back. Because displacement is path-independent, it shares a deep property with the work done by a conservative force such as gravity. That shared property is why potential energy can be defined as a function of position alone.