(covers information from several alternate timelines)
This page contains information regarding Star Trek: Picard, and thus may contain spoilers.
The Federation Starfleet first encountered this vessel in 2266, when a single ship of this type crossed the Romulan Neutral Zone and attacked several border outposts, destroying them utterly. The USS Enterprise responded to the alert and engaged in a tense game of cat-and-mouse with the intruder for more than ten hours, before the Bird-of-Prey was destroyed. (TOS: "Balance of Terror")
The Romulans later used larger numbers to patrol their border with the Federation in force. Typically during an incursion, one Bird-of-Prey would bring a starship out of warp with a plasma torpedo to allow the ships in its battle group to attack the target en masse. When the Enterprise violated the Neutral Zone on the orders of Commodore Stocker in 2267, it was ambushed by a squadron of ten Birds-of-Prey. (TOS: "The Deadly Years")
At least two Romulan Birds-of-Prey had survived until 2380. One was on display as a museum exhibit on Vulcan, where it was used to infiltrate Romulan space for a rescue mission (LD: "Veritas") The other Bird-of-Prey was owned by the warlord Kar Kantar, and was used to attack La Sirena in orbit of the planet Vashti in 2399. The ship, referred to as "antique", was later disabled after combined fire from La Sirena and an unknown attack vessel severed the starboard nacelle. The Bird-of-Prey still managed to score a direct hit on the attacking vessel, which caused it to collide with the planet's defense grid. (PIC: "Absolute Candor")
Grey in color, with its namesake bird design painted on the hull, the Bird-of-Prey was essentially a saucer with parallel warp nacelles mounted port and starboard of the main body. The aft of the ship was designed with a raised "fin."
Prior to the first engagement between this class of vessel against the Federation in 2366, it was asked by USS Enterprise Captain James T. Kirk if this type of vessel could be engaged "with a reasonable possibility of victory", Chief Engineer Montgomery Scott responded, "No question. Their power is simple impulse." (TOS: "Balance of Terror")
Although incapable of matching the Federation's Constitution-class starship one-on-one, the ship still featured technological advances that were previously unattainable to Federation science: the invisibility screen and the massively powerful plasma torpedo. This class of starship had a single forward torpedo launcher. It also carried a number of nuclear weapons. (TOS: "Balance of Terror")
The main bridge featured a rather small command center, providing access to four computer terminals facing each other in a centrally located console. Each terminal had a scope. The terminals maintained the vessel's course control, tactical and cloaking systems. No seating was provided for any of the officers. The bridge had direct access to a corridor in the aft and a viewscreen in the forward section. (TOS: "Balance of Terror")
Ships of the class
In the revised final draft script of "Balance of Terror", the Romulan Bird-of-Prey was at first visually described as "similar" to a Constitution-class saucer section and was further characterized as "definitely [...] some modified version of a starship saucer main section... but with the dark markings on its underside which suggests a bird-of-prey with half-spread wings. And centered there a threatening-looking 'weapon tube' device." Also in the script, the pattern on the vessel's underside was twice likened to a hawk.
This design of starship was originally to have featured in Star Trek III: The Search for Spock, as Kruge's Bird-of-Prey in that film (the first Klingon Bird-of-Prey to ever appear) was at first intended to be a Romulan ship of this configuration.
The Romulan Bird-of-Prey model was designed and built by designer/sculptor Wah Ming Chang and measured approximately 2.5 feet in width. Built in about two weeks, Chang delivered the model to the studio. At the studio however, the model was repainted and the signature "Bird-of-Prey" graphics were applied to the model. 
A full-fledged internally lit studio model, constructed out of vacuum formed plastic, plaster and metal parts, it was then sent to Linwood G. Dunn's visual effects house, Film Effects of Hollywood where the only footage of the model was shot, used in the original version of "Balance of Terror".
It was assumed that the original Romulan Bird-of-Prey model was lost shortly after use, either damaged, destroyed or simply disappeared, causing the D7-class model to be used in its place in the third season episode "The Enterprise Incident". (The Star Trek Compendium, 4th. ed., p. 40), It is, however, concurrently also plausible that the producers wanted to showcase the new model as much as possible, among others as a courtesy to model kit company, Aluminum Model Toys – who, after all, had actually paid for the D7 model – under their exclusivity agreement, in order to give them the most exposure for their new model kit, No. S952.
The whereabouts or fate of the original Bird-of-Prey model have remained somewhat of a mystery, though there are essentially two theories about its fate in Star Trek lore. The first one is that it somehow disappeared. William S. McCullars related on his now defunct IDIC website, how he was told in 1992 by a curatorial assistant of the Smithsonian that the model was not included in their 1992 Star Trek Smithsonian Exhibit, because the private collector who held the model at that time did not want to give it up for public display. The second theory states that the model was destroyed shortly after the episode was filmed, either by mishap on the studio lot or by Chang himself, as he himself had reportedly revealed in a 1982 National Public Radio interview.  The latter resulted from a conflict with the propmaker's union. Chang as a non-member was not allowed to fabricate props for the show nor was he allowed to join. On Producer Robert Justman's urging, who considered Chang's work to be far superior to anything elsewhere available, the studio devised a ruse to make it appear that the props were bought as pre-existing and off-the-shelf from Chang and it was reflected as such in the purchase orders Desilu sent to Chang – in the process also explaining why Chang never received credit for his work. This part of the problems with the union was years later, in 1996, formally confirmed by Justman in his reference book, Inside Star Trek: The Real Story (pp. 119–120). In the case of the Bird-of-Prey, however, the union got hold of evidence that the model was specifically built for the show and agreed to drop the grievance against the studio if Chang did not receive payment for the model. Without going into specifics, Justman referred to this situation when he made the short, evidently terse remark, "A year later, prior to our second season, the same union problem erupted all over again." (p. 120) Desilu gave in and returned the model unpaid to Chang, who, in a fit of anger, smashed the model to bits in his backyard with a sledgehammer. Both theories explain that the model was no longer available by the time the third season of the original series went into production.
A new CGI model version of the model was used in the 2006 remastered Original Series at CBS Digital, where digital animators worked with the model under the supervision of Niel Wray and David Rossi, for representation of the craft in its respective episodes. The model, together with his other Original Series builds, was built by Finnish fan and digital modeler Petri Blomqvist,  and were bought from him by CBS Consumer Products for use in the remastered series. Blomqvist's work was brought to the attention of Wray and Visual Effects Supervisor Michael Okuda by Technical Consultant Gary Kerr. The quality of his work was a compelling reason for the acquisition, as it saved valuable production time. Nevertheless, the digital animators still had their work cut out for them as Blomqvist's model was constructed in the LightWave 3D software, whereas they used the Autodesk Maya CGI software at the time, and had to translate the digital model from one format into the other, which inevitably led to some information loss. Additionally, they had to cut down on the resolution level of Blomqvist's highly detailed high-resolution model, in order to speed up computer rendering time. It, and the other models CBS bought from him for the project, has earned Blomqvist an official "Technical Consultant" credit. (Sci-fi & fantasy modeller, Vol. 26, p. 49-50)
In addition to recreating the original angles of the ship in space, a new shot was included showing the aft three-quarters of the ship, which was previously unseen in the original airing of the episode. A subtle refinement in that shot was the addition of slightly visible hull plating on the dorsal side of the model, that was not there on the original physical studio model. In the remastered series the Bird-of-Prey, aside from the two original episodes, also made a new, additional appearance in "The Enterprise Incident", where one of three D7s was replaced by a Bird-of-Prey.
A prior CGI model was specifically constructed at Foundation Imaging by Robert Bonchune for use in licensed print publications at the behest of Ben Robinson, featuring in 1999 for the first time in Star Trek: The Magazine Volume 1, Issue 4, with subsequent appearances in the 2001 reference book, Star Trek: Starship Spotter, and in the February spread of the 2006 Star Trek: Ships of the Line calendar, reprinted in its book derivatives. Bonchune revisited his build in 2015 by re-rendering it in more detail for its prominent appearance in Robinson's partwork publication, Star Trek: The Official Starships Collection.
In Star Trek Online, this Bird-of-Prey is the starter vessel of Romulan Republic characters. It is classified as a T'liss-class Light Warbird, and is a Lieutenant-level ship. The Bird-of-Prey seen in Star Trek: Picard, known as the "T'liss refit", was made available as part of a limited-time "legendary" pack in June 2020.
- Star Trek: Starship Spotter, November 2001 - vessel specifications
- Star Trek: The Official Starships Collection, issue 57, 2015