Most often, a flour bomb is simply a plastic bag (or other easily-broken container) filled with flour. It is thrown at the intended victim, bursting open and leaving an impermanent stain on their clothes. This is most effective when you are aiming at a politician or other popular figure who is somewhat invested in looking professional (or at least, formal) right this very moment.
The most popularized flour bombs as of this writing are the fairly recent (March 2012) ziplock baggie of flour thrown at Kim Kardashian at The London West Hollywood, where she was promoting her new fragrance, and the 2004 'attack' of Fathers 4 Justice in which they hit Tony Blair with a condom filled with purple flour. Obviously, these attacks are rare and basically harmless, but can be quite highly publicized. Flour bombs are also used as pranks by dumb college kids everywhere.
Flour bombs are sometimes paired with heavier artillery, such as eggs and various liquids, making them even more difficult to get off. Such villainy is beyond the scope of this node.
It is worth noting that airborne flour can explode when ignited, and this might also be called a flour bomb. This can be an interesting and potentially extremely exciting experiment, but should not be used as a weapon, both because blowing things up is wrong, and because blowing yourself up is wrong and stupid.