What makes it so hard to land on an aircraft carrier?
Edit

Since another user, with actual carrier experience, has given a detailed answer, please refer to his answer.

It now appears that a good part of my answer was incorrect.  As I wrote in the beginning, my answer was based on flight-sim experience, and reading, but nothing formal or official.

I'll leave my answer up, solely for historical reference, unless the better answer gets accepted.



I've never done it IRL, but read about it lots, and Flight Simmed it many times, so I'll take a stab.(until someone who has actually done it says better)


Glide Path
When landing on a fixed runway, your glide-path is pretty much right towards the end of the runway.Its not like that at all on a Carrier.  From up high, you are actually aiming for a point in front of the carrier.  You're aiming for where the runway will be in 30 seconds.   As you get closer, the carrier moves closer into position.  But still not entirely in position until the last second.   So you're always aiming for an invisible/imaginary point where the runway will be at the last second.
This is the best picture I've found of what I'm describing:

Notice that the flight path indicator (the little circle in the HUD) is well beyond the front of the carrier.  If this plane was landing on a runway, the flight path would be aimed right at the numbers.
No Flare
In a typical runway landing you flare out just before touchdown, "float" for a second and drop on to the runway.In a carrier landing, you maintain your glide-path attitude all the way on to the deck.   There's no flare or float. This helps you hit exactly the spot you're aiming for (the 3-wire), but in some sense you really are crashing into the deck at a not-inconsiderable descent rate.
Pitching Deck
Remember the glide path to the imaginary touch-down point?  Because the carrier deck can be pitching or rolling in the waves, that imaginary touch-down point is not a fixed distance directly in front of the ship.  As the ship bobs and weaves around, your touchdown point is moving around too.   That makes it even harder to hit the point right.  It can well be at the last second, as your plane is dropping quickly, the deck suddenly jumps up to meet you, making the impact even more violent.
Sideways Movement
The landing part of a carrier deck is the diagonal part on the back of the deck.   It is angled about 20-degrees to the left, relative to the carrier.  But it is moving directly forward with the movement of the carrier.  This means the "runway" is not moving directly away from you, but slipping sideways.