A point of inflection is a point where there is a turning (maximum or minimum) point of the derivative of a graph. This means it is any point where the second derivative is 0 and the third isn’t is a point of inflection and some cases where the the third derivative is 0 can also be a point of inflection (eg the point (0,0) on the graph y=x⁵). This is why points of inflection have an ‘S’ shape as the rate of change of the gradient is changing sign as well as the gradient itself so it change from a curve which is getting steeper to one which is getting shallower or one which is getting shallower to one which is getting steeper.

A point of inflection in the graph of a polynomial of degree 5

Note
While points of inflection are often found at stationary points, particularly when trying to find the nature of the stationary point, it is not a requirement that they are at one.

The number of points of infection in a graph is equal to the number of distinct real roots of the equation formed by equating the second derivative to zero.