Mostly taken from
Starwars.com. This is the quick and dirty version; it explains what was going on with Han and Leia in general, but it doesn't talk about anything else that was happening in the movie, so some things may not make a lot of sense or won't be fully explained. Spoilers, obviously.
Princess Leia Organa is a driven, dedicated individual. She has a forceful, some would say abrasive, personality. She is professional in all respects, but occasionally relaxes her guard to let her strong compassion and quirky sense of humor show. Leia is a petite, fair-skinned human female with brown eyes, and long brown hair that she often wears in an elaborate fashion.
Han Solo is overly confident and arrogant. He possesses a wry, sarcastic wit, and a reckless manner. This often gets him into predicaments that only his foolhardy courage can get him out of. Han is well traveled, and claims to have been from one side of the galaxy to the other. Han is pragmatic, some would say mercenary, and his personality often conflicts with Leia's idealism.
The adopted princess of Alderaan, Leia followed in her adoptive father's footsteps and entered politics. She became the youngest senator ever in the Imperial Senate, representing Alderaan. As a senator, Princess Leia ran a number of "diplomatic aid missions" as fronts for missions for the
Rebel Alliance. She used her diplomatic immunity and her consular ship to gain access to restricted areas. One such mission saw Leia's ship intercepting vital Imperial plans stolen by Rebel spies. These plans, the complete technical readouts of the Death Star battle station, could be exploited by the Alliance to secure a sizable victory over
the Empire.
Han and his
Wookiee first mate and co-pilot Chewbacca had a variety of adventures throughout their long careers as smugglers. Over the years Han and Chewie made countless modifications to their dilapidated Corellian freighter, souping up the Millennium Falcon to become an incredibly fast and well-armed smuggling vessel.
Armed with the Death Star plans, Leia was given a mission by her father to find one of the Republic's greatest heroes and recruit him into the Alliance. En route to the planet Tatooine to find General Obi-Wan Kenobi, Leia's ship was attacked by Imperial forces, and she was taken prisoner by Lord Darth Vader. She barely had time to hide the stolen plans into the memory systems of an astromech droid (Artoo) and ordered it to find Obi-Wan on Tatooine. As a prisoner of Vader, Leia underwent terrible torture aboard the Death Star. In addition to the mind probes, the Imperials forced her to witness the destruction of her home planet of Alderaan. Despite this, she refused to reveal the location of the hidden Rebel base, or the whereabouts of the stolen plans.
During a stretch of bad fortune, Solo was forced to jettison a cargo of contraband spice belonging to the gangster Jabba the Hutt. The gangster put a price on his head so large that bounty hunters across the galaxy would come searching for the Corellian and his Wookiee first mate. Desperate for cash, Solo took a charter transporting transport Obi-Wan Kenobi, Luke Skywalker, and their two droids to the Alderaan system. Upon arriving at Alderaan, they discovered the planet destroyed, and were taken captive by the Imperial Death Star. Eluding the Imperial troops. Han agreed to rescue the captive Princess Leia Organa if he was to be rewarded, and rewarded well. After escaping from the Death Star, Han took the Rebels to their base.
Han decided not to participate in the Alliance's desperate attack against the Death Star, preferring to take his reward and leave. It was in mid-transit that Solo had a change of heart, and joined the battle just in time to cripple
Darth Vader's ship and allow Luke enough time to destroy the station. For his efforts, Han was awarded a medal of honor.
Han received an acting rank of captain in the Alliance, and accompanied the Rebels on numerous missions and adventures, but mindful of his outstanding bounty, refused to join the Alliance officially. He and Luke were fast friends. Leia, however, spent most of her time with Han arguing as both of them tried to mask their rising attraction with irritation and sarcasm. Three years after the Battle of Yavin, when the Imperials attacked the newly-established Rebel base on Hoth, Leia was cut off from her transport vessel and escaped with Han on the Falcon. Despite a number of malfunctions, the Falcon was able to avoid the Imperial Fleet for a time. During quiet moments, when Solo set the Falcon down for repairs, he and Leia grew closer. They shared their first kiss while repairing the Falcon, but their relationship remained far from certain. Han's refusal to commit to the Alliance and constant reminders of his intention to leave the Rebellion to pay off the bounty on his head left Leia reluctant to trust him with her heart.
Unable to repair the ship on his own, Han suggested traveling to Cloud City, whose leader Lando Calrissian, an old friend of Han's, would be able to help. When the Falcon escaped Hoth, Darth Vader had dispatched a number of bounty hunters to recover it. One of them, Boba Fett, tracked the Falcon to Cloud City. As bait for Luke, Leia was to remain with Darth Vader. Han was to be frozen in carbonite and given to Boba Fett so he might collect the existing bounty on the smuggler's head. Leia and Han shared a desperate kiss in the freezing chamber before Imperial troops separated them. Terrified Han wouldn't survive the freezing process, Leia blurted out at the last moment that she loved him, to which he responded, "I know" before descending into the freezing chamber.
With Lando's help, Leia, Chewbacca, and Luke escaped from Bespin and returned to the Alliance but were unable to rescue Han before Boba Fett left the city. The Rebels eventually tracked down Boba Fett as the bounty hunter delivered Han's carbon-frozen form to Jabba the Hutt's palace on the desert planet Tatooine, and planned a rescue mission. Leia posed as a bounty hunter and infiltrated the palace, freeing Solo from his carbonite imprisonment. Han suffered from hibernation sickness, leaving him temporarily blinded and weak. Before they could escape, they were discovered by Jabba. Han was imprisoned and sentenced to death, and Leia was forced to endure the humiliation of being made one of Jabba's slaves. When Luke arrived, he rescued his friends and Leia repaid the gangster by strangling the Hutt with the chain that bound her.
Ready and able at last to commit to the Alliance-- and Leia-- Han returned with the others to the Rebel Fleet where he the commission of General. He was put in charge of a strike team sent to the moon of Endor to destroy the shield generator that protected the second Death Star being constructed in orbit. While on Endor, Leia became separated from the strike team, and befriended an Ewok tribe. Her diplomatic skills aided in convincing the Ewoks to help the Rebel force. It was shortly thereafter that Luke revealed their true relationship to her-- he'd just learned they were siblings, twins separated at birth, and their biological father was the hated Darth Vader.
With the help of the native Ewoks, the Rebel strike force defeated the Imperial forces, and destroyed the generator, paving the way for the victory at the Battle of Endor, a key victory and turning point in the war against the Empire. Han, concerned by Leia's obviously close relationship with Luke, offered to step aside until Leia admitted that Luke was her brother.
After the victory at Endor, Leia continued to demonstrate some ability in the Force, being able to sense her brother's presence. Han and Leia remained close and were married four years after the Battle of Endor.
Han and Leia Pictures (look how cute they are!)
More info:
Leia's BiographyHan's Biography