A point of inflexion is a point where the gradient of the curve stops falling and starts rising, or vice versa. If you imagine driving along the curve, it's a point where your steering wheel would be in the centre position.
Figure 1: A point of inflexion
Another way of thinking about points of inflexion is that they are points where the gradient attains a maximum or a minimum. We therefore locate them by finding those places where the second derivative is zero.