Following Wolf's lead these asteroids were given names associated with the Iliad -- in fact, those in the L4 point are named for Greek heroes of the Iliad, and those at the L5 point are named for the heroes of Troy. Confusingly, the latter group are sometimes called Patroclean asteroids after the most prominent of those, even though Patroclus (the hero) was on the Greek side. However, Patrocles (the asteroid) was the first discovered asteroid at the L5 point, and was named before the Greece/Troy rule was devised.
As the Iliad deals with the events of the Trojan War, the asteroids came to be collectively known as Trojan asteroids. Over time, this term has come to be more generally applied to any planetoidal body at the triangular Lagrangian point of any two bodies -- examples are known to exist in the Sun-Saturn and Sun-Mars systems, among others -- but strictly speaking it applies only to those in the L4 and L5 points of the Sun-Jupiter combination.
There are currently about 1200 known Trojan asteroids, most of them associated with Jupiter, but there are undoubtedly many others too small to be seen with current instruments. 170 had been given permanent names as of October 1999. 624 Hektor is the largest of the Trojans.
E. E. Barnard is now believed to have made the first observation of a Trojan asteroid, in 1904, but the significance of his observation was not noted at the time. It was believed to have been a sighting of the recently discovered Saturnian satellite Phoebe, which was only two arc-minutes away in the sky at the time, or possibly even a star. The identity of the point of light Barnard observed was not realized until an orbit was constructed for the Trojan 1999RM11, an object that was only pinned down in 1999. For failing to realize what he was looking at, Barnard's observation is now only a historical curiosity.