A pawn storm is an attack on an enemy king, led by pawns to attack it. The king is typically castled and you will march your pawns forward, hoping to trade off the oponent's king's pawn cover and blast an open file for rooks. It can be a powerful weapon, and is usually followed up with an attack with pieces. Sometimes you will also use this to sacrfice pawns to open a line of attack or to draw defenders away. Use it, and it will usually work. Sometimes it will get inaffective if the opponent manages to lock up the position and close it up, blocking access. The openings that use pawn storms a lot are the Sicilian Dragon, especially the Yugoslav Attack, where white prepares to attack with the h and g pawns. Hypermodern openings are usually subject to these too, but with a fianchettoed bishop, you may be able to protect yourself. Here is the Yugoslav Attack in action.
I do not mean to say the Sicilian Dragon is bad, in fact, it leads to exciting attacks and counterattacks. Hope you liked it!