Luke Skywalker is the main character of the Star Wars films A New Hope, The Empire Strikes Back and Return of the Jedi. He is the son of Anakin Skywalker and the twin brother of Princess Leia, and in the Legends expanded universe of Star Wars novels, he eventually marries Mara Jade and becomes Grand Master of the restored Jedi Order.
In the films he is played by Mark Hamill. A full character synopsis can be found at Wikipedia.
Luke Skywalker's story is nearly a point-for-point classic portrayal of the monomyth, otherwise known as the Hero's Journey.
He does go through significant character development, transforming from the naive and somewhat whiny kid of A New Hope to the determined hero in The Empire Strikes Back, and finally into the wiser and more stable Jedi of Return of the Jedi.
Overall, he is a nice, sensible kid who handles stress well. He can even be considered a little of a dork where starships and speeders are involved. It's notable that he doesn't lose his good humor and will to fight, even when defeated or in shock. Secretly, he was angry and resentful toward his father (now Darth Vader) until he faced him down before Emperor Palpatine in Return of the Jedi.
There is an awful lot of father/son fluff fic concerning Luke and Anakin. While goodfic can be found, some of it is tiresome, and ends up trivializing the choices of Anakin Skywalker—or even "making it all better" without consistent character development to allow something like that to happen.
In this vein, there's fic about if Anakin never fell and instead raised Luke and Leia with Padmé as a big happy family. Needless to say, if this happened, the entire plot of the movies NEVER happened, and what is left is a big dreadfully dull mess.
In the PPC
For the most part, PPC interactions with Luke are just the usual for canon characters trapped in badfic, though if he is ever replaced, dealing with him is a dangerous proposition, because he is extremely strong in the Force and proficient with a lightsaber. Luke does not know about the PPC, but has at least once played a pivotal role in PPC history, during the Reorganisation. When Nyx Nightingale discovered that the Mysterious Somebody was actually a clone of Joruus C'baoth, and that he would withdraw his mind control of Dassie Hyrax if he sensed another Force-user in Headquarters so as to avoid being discovered, she brought Luke to Headquarters briefly so that Dassie would be freed long enough to identify the MS to the Sunflower Official, leading to the Dark Jedi and his supporters being exiled from the PPC.
In 2008 HST, Luke was in the Department of Fictional Psychology at the start of the Macrovirus Epidemic and trapped in the resulting quarantine. He managed to escape his room, find his lightsaber, and join briefly in the fight against the macroviruses before the nurses caught him and locked him in with Legolas for his own safety.