Campaign progress is stored online. Swapping computers (or reinstalling) shouldn't affect that.
Now that I think about it, I'm not sure if you can save a game in the middle of a mission like you could in SC1, and if so, I'm not sure if that would be saved. But completed missions are completed on other computers.
Update:
I have noticed some things that just don't get saved. Like if you finish a mission, then just purchase armory upgrades, then quit. It doesn't always save the armory upgrades. When the campaign is continued, it picks back up immediately after finishing the previous level. But this happens even on the same computer, and isn't any worse if you reinstall StarCraft II
Blizzard on Starcraft 2 Save Games: Starcraft 2 Blog
The answer is yes, you will be able to install Starcraft II on any future box you have.
Soon after you purchase Starcraft II, you will link your key to your battle.net account. If you don't have a battle.net account yet, you will have to create one. You must link your key to play the game.
Once your account is linked, you never need your key again. What's more, you don't even need the install disc again, as you can download the entire game client off the battle.net site. The disc itself is just a convenient way to install the game; I myself bought it and didn't even use it. The same is now true for any PC Blizzard game released in the past 12 years, including Diablo 2 and the original Starcraft.
What's more, for Starcraft II your match history, achievements, and current campaign progress will also transfer because it's stored on the server. There may be some stuff like Replays which won't transfer over to your new box, but for the most part you will be playing the game where you left off.
The only way you could hose yourself here is by somehow losing access to your battle.net account, for example by forgetting your password or permanently losing your internet connection. I'd recommend against that.
Best Answer
If you still have the Starcraft 2 Disc all you need to do is install it onto the new computer like you did originally.
You do not have to uninstall the copy on your laptop, but you'll be unable to run starcraft 2 from both computers simultaneously (unless you're using offline mode).
If you have lost your starcraft 2 discs, you can install the game via digital download through battle.net, here: