The most vital thing to realise is that water can exist in the liquid droplet state below 0 degrees C, down to about minus 10. This is because water droplets need a nucleus / suitable surface to freeze onto.
If you fly in temperatures below zero, your aircraft will become cold soaked but you are generally safe in clear air, as IO540 said, due to the lack of liquid water (although it is possible for hoar frost to form by deposition).
However, if you then fly your cold airframe through some of this "supercooled" water (i.e. mist, fog, cloud), it provides a nucleus for freezing for any water droplets that come into contact with it.
Airframe icing can form very rapidly, destroying lift, cloaking radio and navaid aerials, coating the windscreen with an opaque layer you can't see through, perhaps also getting into vulnerable flying control mechanisms, possibly preventing flaps (or one flap) from being lowered, for example, in severe cases.
Stay well out of icing conditions; some days it's not just safe to fly a light aircraft (or helicopter) at all. However, generally speaking, provided you comply with the ANO PPL weather limits for VFR (moisture is what reduces the vis, in most cases), you should stay safe. However, it's always a good idea to check the icing level via the Met Office F214 before planning your flight, (or not!).
Note that the metman's definitions of icing (light, medium, moderate, severe) refer to icing accretion (build up) RATES, not total icing. If you fly for long enough in light icing, it can be just as bad as that from a short time in severe icing.