Why do comets grow glowing tails?

Far out in the cold dark of the Solar System, a comet is just a dirty lump of ice and dust only a few kilometres wide, dark and silent. But as it falls in towards the Sun, something extraordinary happens: it grows a glowing head and sprouts a shining tail that can stretch across millions of kilometres of space. And here's the odd part, the tail does not stream out behind the comet like a scarf in the wind. It points away from the Sun, even as the comet speeds off again.