The most skill-based card is probably one in the new set, but based on what we have, these jump to mind. This is all my opinion and I have no stats backing any of this up.
Cards that require a tradeoff for drawing: Cellar, Warehouse, Storeroom
New players buy these blindly and trigger terrible reshuffles with them. I wouldn't say they are high on the list, but they are up there.
Remodel class: Remodel, Upgrade, Develop, Graverobber, Expand, Forge, Butcher
It requires some knowledge of the game to understand when and why to use these, especially like when you want to trash Copper over a more expensive thing or when you want to upgrade a good card instead of a bad one.
Autoplay cards: Throne Room, Golem, King's Court, Procession
Chaining these can get pretty complicated, and understanding what order to play cards in chains and why can be very important.
Cards that care about cards on top of the deck: Wishing Well, Lookout, Scrying Pool, Mystic, Doctor, Journeyman
These require tracking to use effectively, and tracking is a skill you develop with experience.
Duke, Fairgrounds - In my experience, newer players have no clue when to go for this and it's often very powerful.
Tribute - Tribute engines are tricky but totally viable.
Ambassador - Maybe the most skill intensive card in the currently released sets. If you don't understand Ambassador, you will probably lose if you ignore it.
Cards that reorder your deck - Navigator, Apothecary, Cartographer, Scavenger, Scheme
You can set up neat turns, but thinking that far ahead isn't the first thing most people do in a game.
Possession - Totally transforms a lot of games, and there are several nasty tricks you need to learn to get good with this, including how to play against it, which almost requires reading an article on the subject to begin comprehending.
Watchtower - Not the most complex card, but it has so many interactions to learn that it's probably up there.
Bishop - The most misplayed card in Dominion.
Cards you need to buy under a proper condition: Mint, Grand Market, Doctor
Grand Market is probably the biggest game changer, but all of these can require setup turns before pulling the trigger, and the better you are at the game the more easily you'll see how to get there.
Horn of Plenty - One of the more elaborate strategies to learn is the HoP deck.
Knights
Hermit - It's definitely one of the most complicated cards. You have to know how long you want one in your deck, when to go for Madmen, when to USE the Madmen, etc.
Games that shorten the game: Rebuild, Stonemason
Rebuild obviously is its own metagame for the most part, but Stonemason is right up there in the pile-blowing bananza.
Cards that let your opponent choose something: Advisor, Envoy, Contraband
They're all useful under the right circumstances, but playing them blindly is often a terrible plan.
Black Market - Read Stef's article
Governor - One of the most centralizing cards in the game. You have to learn how to play with it and against it, what risks you can take in a Governor game, and so on. Often you have to be careful about leaving as many as 4 Provinces in the supply.
Copper, Silver, Gold, Platinum - Knowing when you should buy Treasure and when you shouldn't is a big deal, and it only becomes a bigger concern as you play higher and higher skill games.
My vote for the most skill-intensive card overall is Stonemason. It's ridiculously powerful, but only if you know what you're doing, and there's a lot to think about when it's on the board.