Creation and design
Lazarus' spaceship was created by an unknown civilization possibly in the far distant past. The ship was designed to hold a single occupant, who would sit in a chair on a raised dais. The chair was on swivel base which allowed the operator to manipulate several small switches and knobs lining the interior of the craft. (TOS: "The Alternative Factor")
The spaceship was never observed in actual flight and indeed did not appear to be designed for conventional space travel. The front of the ship was left open in a wide gap providing the operator access to the interior. Two power panel covers were on either side of the gap through which one would gain access to the ship's power system. The exact operational mode of power was unknown, but the ship could use dilithium crystals as a power source.
The only function that the craft actually demonstrated was to provide access to the negative magnetic corridor, allowing travel to and from an antimatter universe. Its time travel abilities (if the ship was even so equipped) were never demonstrated.
The front gap of the ship contained two rods that, when charged, provided a threshold for crossing into an antimatter universe. If a living being crossed between the rods once activated, the person would dematerialize and be transported to the negative magnetic corridor between the two universes. Once in the corridor, the person transported would walk (or fall) through to the parallel universe. Passing through the rods of the ship would drain the ship's power and require ten to fifteen minutes to re-energize. (TOS: "The Alternative Factor")
Dual universe craft
One confusing aspect of Lazarus's spaceship was an apparent symbiotic relationship it had to an identical craft in the alternate universe. The two craft appeared to be linked in some way and, when in the same place and time yet in different universes, Lazarus and his counterpart could exchange places repeatedly even when nowhere near their own spaceships. It is unknown if this was a result of entering the negative magnetic corridor, an aspect of the ship itself, or a combination of both.
Another linking feature of the two spacecraft was that if one was destroyed in its own universe, the counterpart ship would also be instantly destroyed in the parallel dimension. (TOS: "The Alternative Factor")
Encounter with Lazarus' ship
On stardate 3087.6, the USS Enterprise encountered Lazarus and his ship on an uncharted planet after experiencing a "winking out" of the universe, when, for a moment, everything ceased to exist. After Captain Kirk forced the insane Lazarus into the negative magnetic corridor, his ship was destroyed by phaser fire from the orbiting Enterprise, forever sealing the corridor. (TOS: "The Alternative Factor")
The Star Trek 50th Anniversary Commemorative Poster depicts Lazarus' ship in actual flight (something never seen in the original episode), with the front gap of the ship closed and the ship in flight as an actual spacecraft.