There were a number of ideas floating around about how Blizzard might implement Hero classes (mostly from D&D, in which Prestige classes are the precursor), but what they landed on was an "unlockable class," that's opened up to your account via a quest with a level 80 character-- that quest hasn't been created yet, so we're not sure what it will require*. So, to play as a Death Knight, you get a character of any class to level 80, do the Death Knight quest, and once your character has completed the quest, you can log on, and create a new character (of any race, we're told) that is of the Death Knight class.
That new character then starts at a higher level (anywhere from 55-70, but Blizzard hasn't decided where yet), with starter equipment of the appropriate level (again, Blizzard hasn't decided exactly which equipment yet or how it will work), and then the Death Knight character can work their way up to level 80 as well as a completely different character. Death Knights will have their own Rune resource system, and they will have their own three talent trees, just like all the other classes. It is a completely separate character, unlocked and able to be created when one of your level 80 characters finishes the quest.
*Update: Commenter Avalanche makes a good point: we don't know much about the quest, but we know it will be similar to the Warlock epic mount quest, and that it will not require raiding. Thanks, A.!
Why did Blizzard do it this way, rather than turning your Mage into a Death Knight? Because apparently they said that players felt they were losing something when they simply upgraded-- they want players to play their old character and the Death Knight as well. And what about the lore, and the fact that Death Knights are Scourge-only? And how is it OK that something ingame is conferring the priviledge of creating a new class out of game? Blizzard claims that all will be explained in the quest your character does at level 80-- we'll wait until they finish it to see whether that's true or not. As for why the Death Knights start at a higher level, Blizzard said they didn't want anyone to be walking around as a level 1 Death Knight, killing Kobolds in Northshire. Death Knights (and other Hero classes, supposedly) are meant to start out pretty epic.
Finally, while we're told the Death Knight is the only Hero class that will be included in the Wrath of the Lich King expansion, Blizzard also said that the idea for Hero classes is for the classes from Warcraft III to all become playable.
So that's a 101 on the way Blizzard has chosen to implement Hero classes. I'm kind of disappointed that they didn't choose to upgrade the character-- I would have liked a system that would have upgraded my Mage into an Archmage or my Shaman into a Far Seer. But it's true that this system lets players have their cake and eat it too-- if you want to play a Death Knight, you don't have to have rolled a Paladin or a Warlock, and even after you create the Death Knight, you can still play your original character. As for whether you'd rather play the Death Knight or your old character, we'll have to see when WotLK comes out.