Conservative force

Take the direct route, take a detour, take the scenic loop back to the start. If a force did the same total work whichever route you picked, it is conservative.

The AP Physics 1 CED states the property from both ends. The work done by a conservative force exerted on a system is path independent and only depends on the initial and final configurations of that system, and the work done by a conservative force on a system, or the change in the potential energy of the system, will be zero if the system returns to its initial configuration.

Path independence is exactly what makes a potential energy function possible. If the work between two configurations is a single number regardless of route, that number can be stored as a property of the configuration, and that stored number is $U$. The CED closes the loop from the other side: potential energies are associated only with conservative forces.

The AP examples are the gravitational force, with $\Delta U_g = mg\Delta y$ near a surface and $U_G = -Gm_1m_2/r$ in general, and the ideal spring force, with $U_s = \frac{1}{2}k(\Delta x)^2$.

AP Physics C: Mechanics makes the connection a calculus one in both directions, $\Delta U = -\int \vec{F} \cdot d\vec{r}$ going one way and $F_x = -dU(x)/dx$ coming back.