While, as an adult, serving as the commanding officer of the NX-class Enterprise in 2153, Archer supervised a boyhood Sim – a clone of Enterprise's chief engineer, Charles Tucker III – while the latter remotely flew the the same model in Enterprise's launch bay, during which the craft was also closely watched by the captain's dog, Porthos. Sim found that the model kept pitching slightly to starboard and, though he complied with advice from Archer that he should trim the craft's port aileron, doing so did not stop Sim from accidentally crashing the miniature, just as Archer had done repeatedly before him. This collision caused the breakage of the model's starboard nacelle, but Archer assured Sim that the damage was "nothing a little glue won't remedy." Sim was indeed extremely eager to fix the craft, even moments after learning the eventful news that he was a clone. (ENT: "Similitude")
Jonathan Archer's model spaceship was one of the few ship designs that only showed up as a model and not as an in-universe ship. The script of "Broken Bow" characterizes the model as "a scale model of an early 22nd century Starfleet transport." The vessel's name and number were scripted to have been painted on the model's hull.  However, this is not the case in the final version of the episode.
The remote-controlled toy was one of several props from "Broken Bow" for which illustrators John Eaves and Jim Martin were instructed to create alternative designs. Eaves illustrated a rough concept sketch of the toy as well as a slightly more detailed diagram of the model (along with a concept image of its small zero gravity device), but it was Martin who was ultimately responsible for designing the model.  He based it on the design of the Lockheed Martin X-33. ("Broken Bow" text commentary, ENT Season 1 DVD) (Eaves' conceptual artwork can be seen here.)
The model was built by the Paramount Pictures prop shop. ("Broken Bow" text commentary, ENT Season 1 DVD) It measured 27x33x7 inches and was custom-made out of plastic.  Some shots of Archer's model in "Broken Bow" were visualized with a CGI version of the starship, created by Eden FX. The digital replication had to match its real-life counterpart exactly but it also had several advantages, the most significant of which was that it could fly more convincingly than if the physical model had been suspended from a wire. (Star Trek: The Magazine Volume 2, Issue 10, pp. 28 & 29)
Manny Coto, writer of ENT: "Similitude", was extremely pleased that the model reappeared in that episode, hearkening back to the pilot episode, "Broken Bow". Coto also liked the parallel between a boyhood Archer flying the model in the earlier installment and a similarly-aged Sim flying the same model in "Similitude". ("Similitude" audio commentary, ENT Season 3 DVD)
The practical model of Archer's toy starship showed up in the 40 Years of Star Trek: The Collection auction as Lot #167, described as "JONATHAN ARCHER'S MODEL SPACESHIP". Although it was estimated between US$500 and US$700, it was sold on 5 October 2006 for US$2,200 (US$2,640 including buyer's premium).