"The vent" is a reference to the genital slit on a male dolphin's underside.
The old man is looking to see whether there are sharks nearby that attack the carcass, which would cause a "swirl."
To fillet (or filet) a fish is to bone it and cut it into a long slice:
A trade wind is "any of a consistent system of prevailing winds occupying most of the tropics, constituting the major component of the general circulation of the atmosphere, and blowing northeasterly in the Northern Hemisphere and southeasterly in the Southern Hemisphere" (answers.com).
The dorsal fin is the fin on a fish's back; pectoral fins are fins on the side of his body.
See here for the external anatomy of a blue marlin.
These "sucking fish" are small remoras, fish that attach themselves to other fish for purposes of transport and nutrition. Remoras do not eat or harm the host fish; they eat material dropped by the host, the host's feces, and/or bacteria and parasites found on the host. A remora attaches himself to the host not by means of his mouth but by a specially adapted dorsal fin on the top of his head.
Anthony Burgess notes the "almost religious humility" evidenced by Santiago in the passage quoted above: "It is easy to understand why the novella was, and continues to be, so universally popular. It is about courage maintained in the face of failure. . . . Like the matador with the bull, [Santiago] feels drawn to the magnificent creature, so that, though one has to kill the other, he does not mind who kills whom" (Ernest Hemingway and His World).
In Hemingway in Cuba, Hilary Hemingway and Carlene Brennen claim that this scene was inspired by an actual event from Hemingway's life in the 1930s when he hooked a huge marlin while on board his boat, the Pilar.
Ernest kept on pumping and reeling hard and fast. . . . He pulled back to pump the rod again and the marlin jumped three more times. Everyone on board was in awe at the way this giant rose up and hung in the air, its long, wet body blurring in great twists and bends, then crashing into a wave of white spray. The line sprang back like a banjo string and the beads of water fell from the line. The fish jumped again, this time hanging stiff and high in the air before falling.
Then the marlin tried to sound three times, and each time Papa* held him and brought him back to the surface. By now both man and fish were tired, but the fish had it worse by far. It seemed confused, and made several tight circles while Ernest took in line quickly. . . .
Just then, the great marlin came back to life with a burst of energy. Line screamed off the reel, and the fish came up closer to the port side and flung himself in a somersault of splashing spray. . . . Then straight back from the Pilar the big fish surfaced again, floating higher and resting, flipping its fins slowly above with the waves. The marlin was only a dozen yards away--it was the moment of surrender. . . .
This video includes some great closeups of the mako's teeth, which Hemingway describes in the novella.