Technical details and capabilities
The Fesarius was very large and spherical, with the surface made up of smaller, dome-like features, arranged in a distinct pattern based on triangles. The ship was able to dispatch at least one small craft, described as a "pilot vessel", and could be controlled by just one crewman using a very simple-looking interface.
The vessel was able to scan the entirety of the USS Enterprise in a few seconds, in sufficient detail to then override and shut down some of its systems, a level of control that was described as "extremely sophisticated" and "brilliant." (TOS: "The Corbomite Maneuver")
Neither the Fesarius nor any other vessel of the First Federation has been seen since "The Corbomite Maneuver".
According to the Star Trek: Star Charts (pg. 36), one possible origin for the name was the planetary system Fesarius. This system had a primary and at least one planet, the capital of the First Federation. Fesarius was a K-class star with a magnitude of +3, which was ten times brighter than Sol.
The special visual effects involved in depicting the Fesarius kept "The Corbomite Maneuver" from airing as the first episode of Star Trek. Jerry Sohl, the writer of "The Corbomite Maneuver", explained, "They were having such a difficult time with the special effects for the starship Fesarius... such an enormous ship... and this held them up for so long that it became the tenth episode to be shown." (The Star Trek Interview Book, p. 128)
The Fesarius miniature was designed by Matt Jefferies. (Star Trek Encyclopedia 2nd ed., p. 153) The construction of the model has been attributed to Wah Chang among others on the IMDB site and the Hero Com website. According to Star Trek History the model was a big ball endowed with "a pair of dimmers to make the model ship pulsate with power. The model was covered with sliced ping-pong balls."
When it came time to construct the Fesarius as a CGI model for the Remastered TOS, the visual effects artists tried to stay as close to the original as they could while also adding a level of detail that VFX Line Producer Michael Okuda stated "probably should have been there at the beginning," thus showing, though not overtly, an internal infrastructure beneath each of the ship's many domes. (Spacelift: Transporting Trek into the 21st Century, TOS Season 1 Blu-ray special features)