Kukulkan's starship, original appearance
Kukulkan's starship, with the image of Kukulkan projected
Kukulkan's starship side view with the image of Kukulkan projected
Kukulkan's starship was a highly advanced starship controlled by the winged serpent creature Kukulkan, who controlled the vessel via telepathy. Extremely unorthodox in design, the ship was twice the size of a Constitution-class starship, had a hull composed of crystalline ceramic, and was capable of achieving at least warp 3. The ship was also surrounded by an immense energy field that could project the visage of Kukulkan.
The ship's interior contained Kukulkan's zoo, and the machinery to create virtual worlds in the minds of other cognizant beings. It was also equipped with a powerful probe beam, and transporters that could penetrate deflector shields. Offensive capabilities of the ship included the ability to create a force globe to restrict the movement of its target.
Kulkulkan's ship was encountered by the USS Enterprise in 2270, while investigating an alien probe that had visited and scanned several Federation homeworlds. During the encounter, several members of the Enterprise crew were beamed aboard the vessel.
While on board the ship, the crew of the Enterprise were given the chance to solve the riddle of Kukulkan's city, which Humanity on Earth had failed to solve.
In an attempt to distract Kukulkan, Captain Kirk released a Capellan power-cat. The cat caused damage to the ship severe enough to cause Kukulkan to lose control of it. (TAS: "How Sharper Than a Serpent's Tooth")
Background information
Much was explained in the complexity of this design in the episode's script:
In center of screen, blob of light is nearer and begins to assume a defined shape, that of forward view of what we will see as Kukulkan's ship. It is a highly advanced spacecraft. This forward view almost has the appearance of a demon, with command module as its face, the curving prow as its head, and the propulsion wings as its horns.
The ship is a slender cylindrical vehicle, with tapering winglike propulsion units only vaguely like those of the Enterprise. There is a flattened and sloping command module as its prow; it houses a round, glassy, onyx-shaded globe. Emphasis is on sleekness and engineering sophistication (no portholes, please.)
[On main viewing screen] a ghostly image of Kukulkan's face superimposes itself over the forward view of his ship, which remains visible under image: the prow becomes a snake's head, grinning and gigantic, with jaws agape, huge fangs, and a collar of rainbow feathers. The propulsion wings are closed within image of actual feathered wings. (This is stylized after actual Mayan depiction, please.)
Now we see the advanced spacecraft shrouded inside the ghostly serpent image which conforms to the smaller graceful lines of the ship. Prow is within the head of the image, a fanged dragon's head with a collar of many-colored feathers; the slim hull matches the transparent serpentine body that is covered with multicolored scales; propulsion units near stern are embodies inside wings.