After 1986's Crisis On Infinite Earth reset the DC Universe's continuity and John Byrne specifically rebooted Superman in the pages of Man of Steel, the character was rewritten so that his powers only came to him as an adult and he was very much the last survivor of Krypton. For a while, the Super writers had to be creative with their reintroduction of concepts like Supergirl (an other-dimensional shape shifter), Bizarro (a clone-gone-wrong of Superman) and Superboy (a clone gone right). Superboy first appeared in 1993's Adventures Of Superman #500 along with the other three men flying the Superman symbol after he died at the clawed hands of Doomsday. He was a clone of Superman who was hatched too early in the Cadmus Labs, hence the brash young attitude that exemplified his earliest appearances.
Starting his fictional life as a hip (by mid 90s standards), irreverent clone of Superman, Superboy refused to let anyone call him that. While it seemed at first like Superboy had similar abilities to Superman, it turned out that he actually had a power called tactile telekinesis that essentially allowed him to control an energy field around him that could mimic flight and super strength. This power was an approximation of an aura that the scientists who created Superboy saw around Superman's body while being studied.
After Superman returned from death, Superboy had no problem taking his proper name and moved to Hawaii where he spent some time fighting bad guys and learning to use his powers more effectively in his own title which lasted from 1993-2002. At one point, Superboy's DNA started to unravel itself and the cure, while saving his life, froze his age and appearance at about the 16-year-old level. Soon after, Superboy found himself transported to an island filled with huamoid animals off the coast of Hawaii, but while he was gone all his friends took off. He eventually joined back up with the gang at Cadmus where he was trained and eventually explored something called Hypertime before kicking around the DCU for a while before his book got canceled. It was during this time that Superman bestowed the name Kon-El on him, a traditional Kryptonian name, to let Superboy know that Big Blue considered him family.
Not content to just be a solo act, Superboy has been a part of several teams. The first and probably least well-known was a group of intergalactic party kids called The Ravers in a short-lived book called Superboy And The Ravers (1996-1998). The year that book ended, Superboy wound up on a team with Robin and Impulse called Young Justice that eventually included Wonder Girl and a lot of other young heroes who are important today. That book ended in 2003 and made way for a new era of Teen Titans which featured many of the same characters and also showed Superboy developing actual Kryptonian powers like Superman's in addition to his tactile telekinesis. While a member of the Titans, Superboy discovered that part of the DNA used to create him as a clone was actually Lex Luthor's, a revelation that lead Kon to doubt himself on a regular basis.
That doubt wound up being well-founded as it turned out that Luthor had created Superboy to be a sleeper agent that he took control of before Infinite Crisis. He eventually came back to his senses, but only after beating on his fellow Titans. After that, he refused to be a hero for fear of hurting anyone else only coming back into the game when Superboy Prime and his cohorts threatened the very fabric of reality during Infinite Crisis. Kon-El, who had also been given the name of Conner by his friends, died while valiantly fighting Superboy Prime (2006).
Superboy's death weighed heavy on many of his friends. His girlfriend Wonder Girl and best friend Robin both mourned him in their own ways, which actually included altering their costumes to reflect Conner's. Robin even went so far as to try and re-clone him, to no avail. While sharing their grief, Wonder Girl and Robin even began their own grief-based relationship. Little did they know that Legionairre Brainiac 5 had ordered his teammate Starman to place Connor's body in the same kind of regenerating contraption that brought Superman back to life after his fight with Doomsday. His injuries being so serious, though, it took a thousand years for Superboy to get better, just in time to reappear in the not-really-a-Final Crisis tie-in, but still a good story Legion of 3 Worlds (2008-2009).
Since returning to life and his proper time, Connor has spent his time in Smallville, living with Clark Kent's mom, reuniting with Wonder Girl and Robin and even going up against his other dad Lex Luthor. He dealt with the Black Lantern problem just like pretty much everyone else in the DCU did. While Superman was on New Kandor last year, Superboy hung out with Valor and also with some Legionaries to help during the Last Stand. By Teen Titans #81 Superboy rejoined the Teen Titans and also started starring in his own book again after sharing the spotlight in Adventure Comics with The Legion of Super-Heroes. Now he's still spending most of his time in Smallville going to school—which sounds excruciating for someone of his power set—and fighitng the likes of Parasite and Poison Ivy as well as not knowing what to think about Lori, Lex Luthor's niece. Why any villain cares about Smallville has yet to be revealed.