When the 6th edition of Magic: The Gathering came into existence, so did several large rule changes. The largest of which was known as "the stack".

The stack is really just a different way of playing spells and effects. Whenever something happens you have a choice, you can respond to it by putting something else on top of the stack, or you can pass. When both players have passed in succession the stack starts to resolve. As it resolves you deal with the stack just like it was a stack of real objects, the first one in is the last one to come out (see: lifo).

Here's a good example of the stack in action (taken from http://www.wizards.com/news/pressrelease.asp?19990115a):
I play Hammer of Bogardan on your Fallen Angel. You respond by playing The Hive's ability to create a Wasp token. After The Hive's ability resolves and your Wasp comes into play, you sacrifice it to your Fallen Angel. The Angel, now 5/4, takes 3 damage from the Hammer and survives.

Although this rule annoyed many of the long time players by rewriting some of the rules they had become accustomed to, it also made the learning curve of the game much less steep.