That there is generally a first-turn advantage in dominion is pretty well understood. That it is magnified for some kinds of cards in comparison to others is also fairly well known. But the precise reasons for why the advantage exists and why some cards exhibit them more than others are not. It is my goal to answer these questions here.
I believe that there are essentially two reasons why first-player advantage exists.
The first is the better understood of the two - in a race for depleting piles, the first player is, at some points, going to have had more turns, and thus more chances to get a majority of a particular card (or, in the case of curses, to dish the majority of that card out). This is the case with any card that has the chance of running out before the game ends. Notably, in most every game, provinces. Which is what gives the first-turn advantage for big money (well, ok, duchies sometimes too). But this is more pronounced in games where 'winning the split' on another card. So we see bigger 1st-turn advantage on cards like minion, peddler, hunting party...
Related to this, strategies which go for a three-pile ending tend to have a bigger-than-you'd-have-otherwise first-turn advantage. Moreover, mega-turn strategies have a significant first-player advantage. Apart from the large possibility these have for three pile endings, the first player just gets the chance to 'go off' first. And if the second player 'goes off' before the first player, they must have had fewer preparatory turns, which means that second player will 'go off' smaller.
The second major reason for first-turn advantage is in reshuffle timing. This comes up in player interactions - most normally attacks. Essentially, the deal here is that when player one attacks player two, he attacks 'the same turn' - i.e., if player one attacks on turn 7, he hits player two's turn 7 hand/deck. If player two attacks on turn 7, he hits player one's turn 8 hand/deck. Moreover, if both players would reshuffle between those turns (as is often the case for turn 7/8), that makes a significant difference. If it's a cursing attack, Player two's curse doesn't hit player 1's deck until the 3rd reshuffle, whilst player one's curse hits player 2's deck on the second. Advantage first player.
If, on the other hand, it's a handsize attack, player two is at an advantage, as he's hitting that 3rd-reshuffle hand is more painful than hitting a 2nd-reshuffle hand.
Now, let's look at some concrete, albeit hypothetical examples to try to demonstrate my point a little more clearly.
You both open 2/5 on an IGG board. You both open copper/IGG. Player 1 reshuffles after his second turn, with a deck of 1 IGG, 8 copper, 3 estates. Player 2's deck after the first reshuffle is 1 IGG, 8 copper, 3 estates, and 1 curse, because player 1 dished out that curse before player two was able to reshuffle. Massive first-player advantage.
You both do exactly the same things, opening witch/hamlet. You both play a hamlet on turn 3 and witch on turn 4. Each witch triggers a reshuffle with its draw. But player 1's witch hits player 2 with a curse before that reshuffle, whereas player 2's hits only after.
Hopefully you get the idea.
Now here's some notable cards or types of cards and where the first-player advantage lies with them (and if it's tricky, why). Keep in mind though, that there's an overarching principle that the early turns are more important than the later ones, which adds a little extra wrinkle for first player in general. Also, when I say second player gets an advantage, that's only relative to the 'average case', i.e. what the first-turn advantage would be without these cards. That's almost certainly not going to be enough to overcome the inherent 1st-player advantage in the game. But also keep in mind that 2nd player has his advantages too - the tiebreak rule and being able to adjust his strategy to the opponent. Even so however... I've yet to see a board where 2nd player can really hold his own.
Cutpurse - pretty good 1st-player advantage, because the earlier reshuffles you're more likely to hit a copper on (and it's more likely to matter, too)
Bureaucrat - decent 1st-player advantage, as the earlier reshuffles have more estates in them, and though that gets reversed later on, earlier turns are more important.
Handsize-reducers - second-player advantage, as they attack more powerful hands
Curse-givers - massive 1st-person advantage
Sea Hag - Still 1st-person advantage, for the first major reason (i.e. being able to win the curse race is HUGE), but much less so than other cursers, as it's more likely that second-player forces 1st to discard something good. Plus putting the curse on top of the deck somewhat mitigates the extra-reshuffle-with-a-curse-in-your-deck thing. Not much, but a little.
Jester - well it's board dependent, but I think in general it's second-player advantage. I think the real power of Jester is in grabbing good cards from your opponent (and it also skips them!), and 2nd player will have slightly better chances to grab them, as well as better things to grab.
Bishop, vault - 2nd-player advantage, as when they use it, the opponent's hands are stronger, mitigating the drawback. Embassy goes similarly.
Trash-for-benefit - 1st-player advantage. One major thing these can do is shorten the game, which helps the person in the lead - generally 1st player.