The Thwomp is a spiked stone monster that falls down to try getting the player. Usually found on the ceilings in dungeons, castles, or forts or towers. (which are all kind of the same thing) It falls down quickly and rises backs up slowly. It's just better to avoid them.
Almost all main attacks will not destory a Thwomp, like fireballs, jumping and snowballs. But Mario, Luigi and Toad can Spin Jump (Alt Jump) on it to prevent harm from jumping. Note that it will still go up while you are bouncing on it, so you could go for a ride. Just keep in mind that these creatures will go back up to the ceiling, and could squash you up against the ceiling.