Attack Pattern: They hang in the air, and when you get close, they fly towards the ground, attempting to crush you. In Super Mario Bros. 3 some also attack from side-to-side, and even diagonally.
Basic Fighting Strategies: The easiest thing to do is avoid them. Wait until they are moving back into position - they won't attack while they're doing that, and then move on through.
Advanced Strategies: In Super Mario World, it's possible to spin-stomp a Thwomp without getting hurt (although it won't hurt him either). This may save you a few seconds here and there, but isn't really that useful.
Super Mario Bros. 3 is the only game where you can hurt a Thwomp. Hammers (and presumably invinciblity) will kill him for 100 points.

Attack Pattern: In this game, Thwomp attacks at regular intervals. If you're on the staircase that he's "thwomping", you'll get knocked all the way back down.
Basic Fighting Strategies: You can't hurt Thwomps, all you can do is avoid them. Wait until right after the Thwomp has attacked, and then dash up the staircase as fast as you can. Incidentally, I think that if you're in the air when Thwomp hit the ground, you might be able to avoid falling down the stairs.
Advanced Strategies: You can jump on Thwomp's head in this game. It won't do you any good, and you'll fall off as soon as he hits the ground, but you can do it. Also, if you stand right under him when he attacks, you'll get flattened and slide down the stairs with a funny little animation. Again, this is totally useless, but it's interesting.

Attack Pattern: Thwomps attack in a regular pattern in this game.
Basic Fighting Strategies: Wait until after he attacks, and then move past him.
Advanced Strategies: You can ride on Thwomp's head in this game, and in fact it is an important technique in some areas. To get onboard, just use a back flip when he lands.