It turns out that isolated stars—like our Sun—are rare. Most bight stars are found in pairs, or binary star systems; and it's in these binary star systems that we see Type Ia supernovae, and their relatives, novae.
Let's start where we left off: Imagine a binary system where one star has evolved to a white dwarf, and its companion remains a red giant. The huge expansion of the outer envelope of the red giant allows some of its material to be stolen by the white dwarf through gravity. When the white dwarf grabs enough of this material (mostly hydrogen) it will instantly ignite on its surface in a flash of a fusion reaction. This flash is what we know as a nova. The reaction creating the flash takes place on the surface of the white dwarf and only expels a tiny amount of material.