Well.. it's a bit different from Warcraft 3. I'm guessing you haven't played the original Starcraft from your post. The two biggest changes in gameplay are economy and heroes. In WC3, you have your gold mine, send x peasants in, and you're done. In SC2, you have mineral patches which behave more like forests in WC3 in that there are a bunch of them and you build workers to harvest them - and you need many more workers. The second resource in SC2, Vespene Gas, works kind of like the gold mine in WC3 (build structure, send workers in to extract). Then heroes. In SC2, there aren't any heroes with inventories that level up and gear up with items and stuff like that. It's possible to do with the editor, but the hero units in the campaign are buffed up versions of regular units. So whereas the focus of WC3 gameplay was really around your heroes, in SC2 the focus is mainly on normal units (most missions don't get a hero unit even).
That aside, the campaign is incredibly well done. The missions I've seen so far are very varied, much more so than "build base, kill other base". There are cutscenes and interactions between the major characters after every mission, as well as interactive "screens" not unlike point-and-click adventure games (for example, there's a bar with interactable objects you can click on - Jukebox to change music, TV to watch news about your previous mission, pictures to reminiss about the past, other characters for chit-chat, etc). All of this is voiced and animated. You can also upgrade your units and research new designs between missions.
Basically, the campaign presentation is extremely top notch and better than anything Blizzard has ever done. There are 29 missions in all, 26 needed to "win", and an achievement system that encourages you to replay some missions, as the rewards are pretty neat portraits (this extends to Multiplayer as well, with of course better portraits awarded for MP achievements).