624 Hektor is the largest of the Trojan asteroids, those cloud of asteroids that accompanies Jupiter in its orbit. Although all of these asteroids are usually referred to as Trojans, Hektor is more technically part of the 'Greek camp' -- that is, those asteroids that precede Jupiter, drifting around the planet's L4 point. Hektor is also the namesake of the the 'Hektor family', a group of 12 asteroids that travel together.
In addition to being (currently) the 13th largest asteroid yet documented in our solar system, Hektor is also one of the most elongated; it is approximately 416 km in length and 120-130 km wide1. As it appears to be two-lobed, it may in fact be two asteroids that collided in the distant past. It is a D-type asteroid, and may have an ice core (or cores).
Additionally, it is one of the many asteroids that has its own moon2 -- a smaller, 12 km asteroid named Skamandrios.3 It has been calculated that this formation is likely to be stable over the course of at least a billion years, all else being equal.
Hektor was discovered on February 10, 1907, by August Kopff at Heidelberg Observatory in Germany. Skamandrios was discovered in 2006.
1. source; Wikipedia and some other popular sites have numbers that vary around this range.
2. In the last 30 years we have discovered that it is fairly common for asteroids to have orbiting companions, and have found hundreds of examples.
3. The Trojan hero Hector named his son Scamandrius. In a sad twist of nomenclature 175 Andromache, named after Hector's wife Andromache, is far from these two, living in the asteroid belt.