Short answers, pulled from the story.
A trojan is a small celestial body, mostly an asteroid, that shares the orbit of a larger body while remaining roughly 60 degrees ahead of or behind it near one of its Lagrange points. The gravitational balance at these points allows the trojan to orbit with the same period as the larger body without drifting away.
The first trojan asteroid, 588 Achilles, was discovered on the 12th of February 1906 by Max Wolf. Carl Charlier recognized that the asteroid was caught at Jupiter's Lagrange point, confirming in practice the theoretical calculations Joseph-Louis Lagrange had worked out in 1772.
More than 7,000 Jupiter trojans are currently catalogued, but astronomers estimate that more than a million Jupiter trojans larger than one kilometer actually exist. They are divided into the Greek camp at the L4 Lagrange point and the Trojan camp at L5, and their total number is thought to rival that of the main asteroid belt.
By convention, Jupiter trojans are named for figures from the Trojan War of Greek mythology. Asteroids at the L4 point carry names of Greek warriors, while those at L5 carry Trojan names. Two exceptions predate the convention: 624 Hektor sits in the Greek L4 group, and 617 Patroclus sits in the Trojan L5 group.
Jupiter, Mars, Neptune, Uranus, Earth, Saturn, and Venus all have known trojan objects. Jupiter has the most with more than 7,000 catalogued; Mars has nine, Neptune has 31, Uranus and Earth each have two, and Saturn has one known trojan, 2019 UO14. Venus has a temporary trojan. Mercury has none.
As a rule of thumb, a trojan system is likely to be long-lived when the star's mass is at least 100 times the planet's mass, and the planet's mass is at least 100 times the trojan's mass. The formal stability condition for circular orbits requires that 27 times the sum of the three pairwise mass products be less than the square of the total system mass.