The gameplay conceit behind the Carpenting minigame is that you're attempting to use differently-shaped planks of wood to cover gaping holes in the body of your vessel. Unlike the Bilging minigame, there is a time-sensitivity to this -- leave a hole open too long, and it will get bigger.
Your tools for this job are simply the planks of wood, the shapes of which you can see in the photo that accompanies this post. It's a good thing that the tutorial states outright that it's impossible to perfectly place every plank, because frankly that's asking too much of the average human mind. Instead, you're asked to do the best job you can by the overlapping of planks, with as little waste as possible. Doing a good job gets you a pat on the back, and you get closer to finishing the task.
The first time you start out, you get one hole, and a group of pieces that make it fairly obvious how to use them all to fill the hole perfectly. Once you've finished, you move on to four holes at the same time, and it's here that you will truly experience the sensation of life as a pirate as you begin to swear aloud, using terms you didn't realize you knew. See, although there may not be an actual clock, one you've placed a plank in a hole, it's not too long before it begins to glow red and shake, meaning that you have 3 more plank placements before its corresponding hole widens even further. Over time, this small poke at your spatial awareness sense, multiplied by four, begins to unnerve even the most stalwart seafarer.
But it gets even better. Were it not enough to be frustrated by your own inability to do a neat bit of carpentry, there's a bonus to be had for matching the horizontal woodgrain of all the planks, for which you receive a hearty 'Craftmanship!' message. Not only did this fail to inspire me to do it again, I didn't even realize I'd done it in the first place. Frantically dividing my attention between 4 empty holes and the middle section which holds the planks, there was no way in Davy Jones' Locker that I would have been able to match woodgrain, much less keep from screaming with abandon each time the dreaded red alert began pulsing its deadly warning.
So, what's the key to doing well at this minigame? I recommend one of two approaches:
1) Train with the Ural Mountains anchorites until your spatial awareness is honed to a fine point, to such an extent that you can instantly and viscerally sense the proper orientation and location of each plank as it should be laid.
2) Do heroic amounts of mescaline. If you don't begin to see the patterns then, you never will.
(Disclaimer: Massively neither encourages nor condones the use of drugs for the purpose of achieving better gameplay. Winners don't use drugs, kids! William S. Sessions, the Director of the FBI says so!)