The ice cream headache is sometimes called the cold stimulus or brain freeze, but its scientific name is sphenopalatine ganglioneuralgia, which basically means a nerve pain in a biological tissue (ganglion) found in the head and neck.
If something cold touches the roof of your mouth, it causes the blood vessels in your sinuses to cool rapidly and then return their normal temperature. The cold makes them constrict (shrink into themselves) and then they rapidly expand when they warm up again. The pain receptors near the area sense the dilation and this activates a facial nerve called the trigeminal nerve. As it is a facial nerve, the brain interprets the pain as coming from the face - the forehead, in fact.

It is also thought that the pain may be caused by increased blood flow to the frontal lobes of the brain caused by the rapid cooling of the palate.