Heroic is a term used to described the difficulty level of instances and raids, as well as the size of a given raid instances.The Burning Crusade, every new 5 man instance in the game had an Heroic Mode added to it. In order to do the Heroic mode instance, you needed to first hit level 70, then get enough Reputation to hit Honored. At that point you'd be able to buy the Heroic Key in order to engage the Instance on Heroic Mode.
Wrath of the Lich King continued this, but went a few steps further. Not only did every 5 man instance have a Heroic Mode - so did raids. A key difference however is that Normal Raid are designed a raid group of 10, while Heroic raids are designed for 25, allowing for much more different and complex challenges. The Reputation requirement was also lifted, all that is needed to enter an Heroic now is to be level 80.
Switching the Instance between Normal and Heroic mode is a simple matter of the leader of the party/raid clicking on his portrait, going to 'Dungeon Difficulty' and clicking the desired difficulty. The instance will than switch for everyone in the group.
The most noticable thing that happens when the difficulty is changed to Heroic is that the instance becomes harder. Mobs will have more hp, they will hit harder and many, especially bosses, will have brand new abilities they did not possess in the Normal version. This all goes toward making the instance much more challenging.
Heroic raid are also designed for 25 players, where the normal version are for 10. The jump in difficulty - especially the monster's hp and damage - is even greater here.
They are many reason to do Heroic however.