As a Y!PP pirate of about a year, I feel compelled to at least expand the puzzle list up there. (Disclaimer: Puzzle Pirates, not unlike, say, StepMania, exists in that sort of indefinite post-beta phase. I can't speak for other MMORPGs, but the game structure is volatile, and new puzzles can pop up at any new release. The core of the game is more-or-less constant, though.)

Sailing: Yes, it's like Dr. Mario, and yes, rows/columns of 4 score. (I use "score" loosely - one of the overriding policies of Y!PP is "no numbers". Experience and standing are given by vague terms, not absolute stats, for example.) The important part, however, is the target platforms. Patterns of the 3 colors will appear - a triangle, for example, with one color at the hypotenuse, one at the right angle and one between the midpoints, or a 2x2 in a checkerboard pattern - and these affect the performance of the ship the most. Targets are also the only thing that contribute to navigation in battle.

Bilge pumping: Is really more similar to Tetris Attack, just with reverse gravity. You can only switch horizontally, but you're not constricted to switching toward a 3-in-a-row. There are also 3 special blocks - a crab that stays put until it rises above the water line (which rises and falls as the bilge level itself does), a puffer fish that takes out a 3x3 group of blocks, and a jellyfish that takes out every block that matches the one you switched it with.

Navigation (non-battle): Not only increases the effectiveness of the other ship puzzles, but allows the navigator to permanently memorize the paths between islands (towns, as it were).

Swordfighting: "The type of blocks sent..." is kind of misleading. The shape and size of what you send is merely dependent on what you break. Unfused blocks send over half the volume you broke; fused blocks send over the same volume, and nearly always in the same shape you broke. (Gravity-type combos double, triple, etc. the effect.) The colors that make up that garbage depend on the sword you wield - more expensive swords have more intricate patterns that don't lend themselves to providing easy-to-clear garbage. (For example, the foil - the default, green, fresh off the island sword, pretty much sends 3 columns of solid color, with the occasional 4th color to make things interesting.)

Rumble: An alternate to swordfighting. More or less a 2-base version of Puzzle Bobble or Snood or all those marble bouncing games. Different weapons ("bludgeons") affect attacks the same way different swords do. The addition: if you hold your firing button for a second, then hit a cluster that would break otherwise, the cluster won't break and the fired marble will stay glowing. Do a bunch of these and break the first one for a better combo.

Drinking: Somewhat recently had "mugs" added, which give bonuses to the players depending on certain events. For example, extra turns when you complete a row, or points when an opponent completes a row you were part of.

Card games: Three other recent additions, all played the same as their meatspace equivalents. Hearts is Hearts, Spades is Spades and Poker is Texas Hold 'Em.

Crafting: There are now 3 crafting puzzles, and the hope is for a puzzle for every shop. Distillery ("brewing") is pretty much as m_turner said. Shipwrighting involves switching blocks on a 5x5 grid to match a set of patterns associated with different ship parts, and Alchemistry (not to be confused with PopCap's "Alchemy") involves manipulating a set of pipe segments to make a path between tanks of potions (in each primary color, at least as was taught in elementary school) and empty potion jars at the bottom.

Other minutiae:

There are two kinds of servers, or Oceans. Subscriber oceans cost US$10 per month, with no limits; Doubloon oceans cost no commitment to buy, but use a micropayment system (the doubloons) on top of the normal currency.

The nature of the trial period has changed. Originally unlimited puzzling (with exception of gunning and navigation, which are only available to certain grades of crewmembers, of which non-subscribers cannot be a part) but only 10 logins available, Y!PP now allows for unlimited logins, but limited puzzling, only allowing one or two days a week per puzzle. For example, outside of battle, you can only swordfight on Wednesday, Friday and Sunday.