This whole writeup is a spoiler, since the nature of the Galactic Leyline is not revealed until the very end of the series.
The Galactic Leyline is the artifact of an ancient civilization of aliens, and the great prize in the anime Outlaw Star. The Leyline is, among other things, a vast library which contains everything the civilization knew. In addition, it has the power to utilize the knowledge itself, granting any wish to those who can find and enter it. It's powers are so great that it is virtually omnipotent (Melfina refers to it as a god machine). For example, the Leyline was able to bring Gene, Melfina, and Hazanko back to life after they were killed in an explosion. Obviously, gaining access to the Leyline is something that is high on the priority list for a large number of people, particularly Hazanko.
Knowledge of the Galactic Leyline is quite limited, outside of the highest ranks of the pirate groups (and possibly in the top levels of the human and Ctarl-Ctarl empires). For example, Aisha Clan Clan believed that the
Leyline was a massive treasure house, full of gold or dragonite.
The location of the Galactic Leyline is particularly controlled; the only person who knew it who wasn't a pirate was a guy serving life on a prison planet. But the location alone is useless; the Leyline is surrounded by a strange field, which according to all sensors is empty space. But anyone who enters this field will quickly go insane (as the crew of a Ctarl-Ctarl battleship found out to their misfortune). When the Outlaw Star arrived at the Leyline, they found it surrounded by pirate ships, including the Geomancer, who were attempting to force an entrance into the Leyline.
Melfina, a biological android created by Gwen Kahn, is the key to activating the Leyline. Included in her genetic makeup is some DNA found on a planet formerly inhabited by the ancient civilization. Dr. Kahn was able to
decipher parts of the DNA that was found, and he used the information it contained to design the Outlaw Star, which is why it can safely enter the Leyline. But the remainder could not be analyzed, and Kahn thought that the only way to activate the Leyline would be to use Melfina as a "container" for this information.
Melfina was, in fact, able to guide the Outlaw Star through the protective barrier and into the Leyline. Hazanko, in the Geomancer, and the McDougall brothers, in the El Dorado, were both able to follow in the Outlaw Star's wake and enter the Leyline as well. Once they left their ships, the Leyline transported Melfina into the center of a maze, leaving everyone else scattered at the edges. As each crew attempted to reach the center first, either to claim the prize of the Leyline, or rescue Melfina (or both), they began to fight it out amongst each other, resulting in the death of damn near everybody (but, as mentioned previously, the Leyline decided to bring several of them back, and a few supposed deaths, including Aisha's, were just cliffhangers).
After Hazanko was defeated by the Outlaw Star, the Leyline decided that it was time to move to a new location, and disappeared, presumably transporting itself to somewhere else in the galaxy.