A feasible point $x_*$ is a local minimum when nearby feasible points do not have smaller objective value. It is a global minimum when no feasible point anywhere has smaller value.

Local and global optima are different concepts: a function can have several local minima with different objective values.

For an unconstrained differentiable function, an interior local optimum must satisfy the first-order condition

$$\nabla f(x_*)=0.$$

Such a point is called a stationary point or critical point.

The condition is necessary under ordinary smoothness assumptions, but not sufficient. A stationary point can be a local minimum, a local maximum or a saddle point.

Optimization algorithms often search for stationary points because local derivative information can identify them even when finding a guaranteed global optimum is much harder.