Hot spots form when heat from deep within the earth causes mantle rock directly beneath the crust to melt and form a magma plume. This occurs because of the heat and because the pressure directly beneath the crust is relatively low, allowing melting to occur. It is important to note that this process is purely theoretical, as scientists have not yet observed it directly.
Hot spots are regions of anomalous volcanism, or volcanism which occurs away from a plate boundary. At plate boundaries, magma seeps up from the mantle relatively easily, which is the reason for the Pacific Ocean's Ring of Fire, the ring of volcanoes surrounding the Pacific plate. However, hot spots result from a different process, and result in chains of volcanoes far from a plate boundary.
One notable example of hot-spot volcanoes is the Hawaiian island chain. Each of these islands is actually a volcano that formed above a hot spot in the Pacific plate. The chain results from the Pacific plate's movement relative to the mantle hot spot, so new eruptions take place at a distance from the previously formed volcano. While the Hawaiian islands are well-known examples of hot-spot volcanoes over oceanic plates, hot spots occur under continental plates as well.