What you are seeing is the air glowing. The sun is always sending out tiny invisible particles into space. When some of them reach the Earth, the Earth's magnetic field guides them down to the North and South poles. There, they smash into the gases in our air and make the gases glow - the same way electricity makes a neon sign light up.
Different gases make different colours. Oxygen, very high up, makes red. Oxygen lower down makes the famous bright green. Nitrogen makes pink and purple. On a great night, the sky can flicker through several colours, all at the same time, in shapes that look like dancing ribbons.
To see them, you need three things: a dark sky (away from town lights), a clear night (no clouds), and patience. Many Icelandic families have a tradition of bundling up in warm coats and waiting on a quiet hill, with hot chocolate, just watching the sky.
The Northern Lights move silently, but people often say it feels like they should make a sound. In some Icelandic stories long ago, the lights were thought to be the spirits of ancestors waving from the sky. Today we know what makes them - and they are no less amazing for it.