From Webster's Ninth New Collegiate Dictionary:
dog n. ... 3 a : any of various usually simple mechanical devices for holding, gripping, or fastening that consist of a spike, rod, or bar
Alas, that definition doesn't help much.
I suppose a dog clutch is a "clutching mechanism that employs a clutch dog. That's no better, as it seems to be self-referential.
The better answer is "the mechanism composed of item #38 and the adjacent gears #32 and #33 in this exploded parts diagram (143k)." (This page scanned from the OMC Cobra Parts Catalog.)
Item #38 is the "clutch dog" inside a Cobra sterndrive. It's a stout, hollow, sliding cylinder splined to the propeller shaft, with teeth cut into each face for engaging the adjacent gears.
The teeth on the clutch dog resemble the decoration on top of a chess-piece called the "rook" (or "castle").
Immediately adjacent to the clutch dog on the propeller shaft are gears that spin freely on the shaft. These gears mate with a third gear that is driven by the engine, thus these gears are always turning while the engine is running. These gears turn in opposite directions.
The sterndrive is in neutral when the clutch dog is between the nearby gears; the drive engages forward or reverse by sliding the clutch dog into the appropriate gear that turns with the engine.
There is no "clutch," as you find in an automobile. Gears engage with a jolt. Some say "the water is your clutch," but I always thought a clutch had to be between the engine and the load to be useful ;-) .
#39 is the pin that forces the clutch dog to turn with the propeller shaft. #18 is the propeller shaft, and #15 slides fore/aft to drive the clutch dog.
Two different gears are marked #32; the one on the propeller shaft is the forward gear, and the other turns with vertical driveshaft #35. Gear #33 is reverse.
Shifting is accomplished by raising and lowering rod #36; it connects with detented arm #11. Ball bearing #6 (pressured by spring #12) falls into the detents milled into #11. Arm #11 rocks bellcrank #14 to and fro, transforming the up/down of #11 into fore/aft on cradle #13 and shifting shaft #15.