Kyp Durron is a human Jedi of Luke Skywalker's new Order, introduced in Kevin J. Anderson's Jedi Search (Bantam, 1994), the first volume of the Jedi Academy Trilogy. He is a slave labourer in the Imperial spice mines of Kessel when Han Solo and Chewbacca meet him, and one of the strongest raw Force talents Luke has found.
The Sun Crusher
What follows is the most notorious plot in Bantam-era Star Wars. Possessed by the spirit of the ancient Sith Lord Exar Kun, Kyp takes the superweapon prototype called the Sun Crusher and uses it to destroy the Carida system — an Imperial military world, and a populated one. His own brother is among the dead. Estimates of the toll run to millions.
The problem the Expanded Universe never solved
He is then redeemed. He helps defeat Exar Kun, he is forgiven by Luke, he faces no trial, and across the next decade of novels he rises to Jedi Master and commands his own squadron. Whether that redemption is earned is the longest-running argument in Legends fandom, and the books themselves are divided: Michael Stackpole's I, Jedi (1998) exists partly to voice the objection, through Corran Horn, that the Jedi handled Kyp Durron appallingly.
Kyp's Dozen, and the Vong
The New Jedi Order series gave him the best material of his career precisely because it refused to smooth him out. He runs an independent squadron, he takes pre-emptive action against the Yuuzhan Vong without the Order's sanction, and he is directly responsible for provoking at least one catastrophe. He is the argument, inside the fiction, for what happens when Jedi decide they know better than the government they serve.
Reading him
The Jedi Academy Trilogy for the crime, I, Jedi for the prosecution, and the New Jedi Order run — Dark Tide through The Unifying Force — for the man who has to live with it. He appears in the Legacy of the Force and Fate of the Jedi series too, still arguing.
Why he matters here
Because he is the load test for Luke Skywalker's whole project. Luke rebuilt the Jedi from scraps of a religion nobody had practised in a generation, with no Council, no oversight and no code but his own judgement — and Kyp Durron is what that produced on its first attempt.
Continuity
He is Legends. Nothing in current canon has replaced him, and canon has so far declined to give Luke's Order anyone remotely as compromised.