Romulan Bird of Prey
In Star Trek, the designation "Bird of Prey" was originally intended to describe only Romulan starships, most famously the kind seen in Balance of Terror (TOS). The Romulan Bird of Prey was a saucer shaped ship with two upswept wings with warp nacelles at the ends and its namesake bird painted on the underside. Its main weapon was a forward firing plasma torpedo launcher, but its most distinguishing feature was the cloaking device.
Due to the budget constraints of the original 1960s show, however, the Romulans and the Klingons were often shown to use similar starships and weapons. This was explained by a trade agreement between the Klingon Empire and the Romulan Star Empire. The most notable results of this trade were the Klingons receiving cloaking technology and the Romulans receiving larger, more capable warships such as the Klingon D-7 class battlecruiser (as seen in The Enterprise Incident (TOS) ). The Romulans were later seen in The Next Generation to have developed their own large warship in the D'deridex Class Romulan Warbird (The Neutral Zone (TNG) ).
The trade agreement muddied the waters a little between what was Romulan technology and what was Klingon technology. Combined with the fact that (aside from the Klingons being more aggressive and the Romulans being more subtle) they were essentially treated as interchangeable enemies of the Federation; mistakes were bound to be made.
Klingon Bird of Prey
According to Memory Alpha (http://memory-alpha.org/), such an error occurred in the production of Star Trek III: The Search for Spock. Originally, the Klingon ship was supposed to be a Romulan ship stolen by the Klingons. Although that element of the plot was dropped from the final revision, the ship was still referred to as a Bird of Prey, and still had the upswept wing design typical of Romulan ships as opposed to the downswept wings of Klingon designs. This was also the first Klingon ship seen to have a cloaking device.
When the Enterprise crew captured this "Klingon Bird of Prey", it wound up being their vessel for the next movie, Star Trek IV: The Voyage Home (aka the one with the whales). Precedent was set, and pretty much any Klingon ship smaller than a battlecruiser adopted the moniker Bird of Prey as well. Klingon Birds of Prey were agile ships with forward facing disruptor cannons on the end of each of the wings and a photon torpedo launcher in the front, under the bridge. The design was similar to the Klingon battlecruiser, with a long neck connecting the bridge to the primary hull, which had the downswept wings.
Star Trek: Enterprise
Then Star Trek: Enterprise, the prequel to the Original Series episodes, muddied things up even further by introducing cloaking technology (Minefield (ENT) ) and Klingon ships referred to as Birds of Prey (Borderlands (ENT) ) decades before they were supposed to have existed.
Ultimately, it's best not to think about it too hard. The Klingons and Romulans both have Birds of Prey, and both have cloaking devices. When and why exactly are not clearly established, and continuity problems abound.
Sources:
Memory Alpha (http://memory-alpha.org)
and my recollection of the relevant shows and movies
Thanks also to Jack for some additional info!