A Sun-Eater is the name commonly attributed to a weapon created to destroy stars. Guided by some inexplicable instinct, it seeks out vibrant stars and envelops them, absorbing all of their solar radiation. The star in turn, compresses in size, but not in mass. This results in the dying star going supernova. The thermal energy released from the nova-blast powers the Sun-Eater and propels it towards its next destination.
The alien race known as the Controllers developed the Sun-Eaters as a way to destroy entire worlds that they deemed to be too "evil." Each Sun-Eater was kept in a dormant state until needed, watched over by a Controller. Many years ago, the inter-stellar warlord known as Mongul learned of the Sun-Eater and killed the Controller in charge of watching over it. With the power of the Sun-Eater at his command, he turned it against Earth, the ultimate revenge against his most hated foe - Superman. Fortunately for Superman, he found a helping hand in the form of a group of time-traveling heroes from the 30th century known as the Legion of Super-Heroes. While Superman fought Mongul, Legion member Wildfire apparently sacrificed himself by exploding his anti-energy body inside the Sun-Eater's core, but he managed to re-form. 
In the 30th century, the Controllers developed another Sun-Eater and placed it in the care of a specified keeper. The keeper of the Sun-Eater eventually became insane and released the weapon on the Milky Way galaxy unprovoked. To stop the star-sized creature, the Legion of Super-Heroes recruited some of the worst criminals in the galaxy to help them (these villains would later go on to form the Fatal Five).  In the end, there was only one solution towards stopping the Sun-Eater. An "Anti-Energy" bomb made of Absorbium would have to be detonated inside of the Sun-Eater's core. Only Superboy was invulnerable enough to deliver the bomb inside, but he was weakened by red solar radiation from inside the Sun-Eater (from the red suns it had already consumed.) New Legion member Ferro Lad, who possessed the power to turn into living iron, could resist going inside the Sun-Eater but not the bomb's explosion. Heroically, he stole the bomb and delivered it anyway, killing himself in the process, but destroying the Sun-Eater, thereby saving the galaxy. 
Several years ago, a Sun-Eater made its way into Earth's solar system and consumed the sun. Earth's heroes knew that the world had less than a week before the lack of sunlight would make the planet incapable of sustaining life. Initially, they attempted to alter the Sun-Eater's initial trajectory, by funneling it through an over-sized Boom Tube, but this strategy ultimately failed. They also attempted to lure the Sun-Eater away with an artificially constructed secondary sun as bait. This too met with unsatisfactory results. Despite all of their vaunted efforts, the Sun-Eater still succeeded in consuming the sun.  The being known as Parallax flew towards the sun and used all of the power at his disposal to absorb the energy of the Sun-Eater into his own body. This resulted in the destruction of the Sun-Eater, but also the destruction of Parallax as well. When Parallax died, his body eschewed all of its stored energy to reignite the sun. 
More recently, the Guardians of the Universe came into possession of a miniature Sun-Eater and are currently using it as a means of imprisoning the mad Kryptonian, Superboy-Prime. Red solar radiation robs a Kryptonian of their super-strength, and the stored radiation present inside the Sun-Eater is adequate enough to keep Superboy contained. 
Recent circumstances also reveal that it may be possible that some Sun-Eaters even exist as a sentient form of life. Assuming the shape of large translucent brains, some Sun-Eaters have been shown to demonstrate certain animalistic traits such as instinctual navigation and migratory patterns. The super-hero known as Animal Man was once able to tap into the "brain" of a herd of Sun-Eaters and simulate some of their properties. 
In one instance, a Sun-Eater was destroyed using a highly advanced "Entropy Bomb". The Sun-Eater had attacked an unidentified star system, and an alien pilot attempted to launch the weapon, but the Sun-Eater attacked him, nearly destroying his ship. Fortunately, Superman was within hailing distance, and carried the bomb into the heart of the Sun-Eater and destroyed it. 
- The exact dimensions of a Sun-Eater are unknown, but it has been established that a fully-developed Sun-Eater has a diameter in excess of 2,000 miles.
- In the non-continuity title All-Star Superman, a baby Sun Eater is part of the intergalactic zoo in Superman's Fortress of Solitude. It feeds off miniature suns, created by Superman with a cosmic anvil from New Olympus.
- Chronologically, the Legion of Super-Heroes' experience with the Sun-Eater in the 20th century took place after their experiences with the Sun-Eater in the 30th century, and so when they met Superman, they already possessed knowledge of a Sun-Eater's full potential.