(covers information from several alternate timelines)
Lieutenant Commander Jack R. Crusher was a Starfleet officer. Considered by Jean-Luc Picard to have been his best friend, he served under Picard's command on the USS Stargazer. He was husband to Beverly Crusher and father to Wesley Crusher. He was killed on an away mission in 2353. (TNG: "Encounter at Farpoint", "Family", "True Q")
Jack was introduced to Beverly sometime in the 2340s by mutual friend Walker Keel. (TNG: "Conspiracy") It took months for Beverly to figure out her attraction to Jack. (TNG: "The Price") He proposed to her in 2348 through a gag gift, a book entitled How to Advance Your Career Through Marriage. (TNG: "Family")
Their son, Wesley, was born in 2348. Jack recorded a holographic message to Wesley shortly after his birth, intending it as a present for when Wesley reached his eighteenth birthday. Jack wanted to make a series of messages, but by the time Wesley was five years old, had only completed the one. (TNG: "Family")
The family took a camping trip to Balfour Lake sometime before his death in 2353. During the trip, Wesley amused himself by throwing manta leaves into the fire and watching them pop. This prompted a response from Jack. Beverly was telling this story to Picard in 2370, but was interrupted before she could tell him what Jack said. (TNG: "Attached")
The Crushers spent much time with Jean-Luc Picard, resulting in Picard developing an attraction towards Beverly. He never acted on his feelings even after Jack's death, feeling to do so would betray his friend. (TNG: "Attached") In an alternate future timeline, Jean-Luc and Beverly did get married but subsequently divorced. (TNG: "All Good Things...")
Crusher was killed on an away mission while serving on the Stargazer. (TNG: "Encounter at Farpoint", "The Bonding") His death was considered an accident. (TNG: "Attached") It also involved a choice of life and death made by Picard, which resulted in Wesley having to make a similar choice at his "psych test" during his first application to Starfleet Academy. (TNG: "Coming of Age") Picard personally returned Jack's body to Beverly and Wesley (TNG: "Encounter at Farpoint"). Picard accompanied Beverly to view Jack in the morgue on Starbase 32. When Beverly was mentally assaulted by the Ullian Jev in 2368, she was forced to revisit this event. (TNG: "Violations")
In 2367, Beverly brought several of Jack's personal belongings out of storage and aboard the USS Enterprise-D, including How to Advance Your Career Through Marriage and the computer chip containing the holographic recording. She gave it to Wesley and he watched the recording on the holodeck. (TNG: "Family")
While on a vision quest on the Native American world of Dorvan V, Wesley saw his father. The vision told him that the quest which Wesley had been on – following Jack's death – was now ended, and that Wesley must now find his own path. (TNG: "Journey's End")
- Early 2340s: Introduced to Beverly Howard by Walker Keel.
- 2348: Marries medical student Beverly Howard. (TNG: "Family")
- 2348: Wesley Crusher born. Jack records a message to Wesley, which will be replayed nineteen years later. (TNG: "Family")
- 2353: Dies on an away mission while serving on the USS Stargazer. Jean-Luc Picard returns his body to Beverly and Wesley. (TNG: "Encounter at Farpoint", "Violations", "True Q")
Jack Crusher was originally to have appeared in the first season episode "The Battle". According to the first draft of that episode's script, he was to be pictured in a still hologram of himself, his wife, and Picard that had been taken the day before the Stargazer launched. All mention of Jack Crusher was removed from the script for "The Battle" by the time its final draft was issued. 
The novel Reunion provides details about Jack's death; he was conducting external repairs of the Stargazer with security chief "Pug" Joseph, but in the process Joseph basically had a panic attack and stayed behind while Jack carried out the repairs and was injured, Pug feigning unconsciousness to create the impression he was injured as well and unprepared for Picard to take him inside before Jack as the other two were closer friends. Pug spends years blaming himself for Crusher's death, feeling that the other man might have lived if Picard went for him first, but when he admits the truth to Beverly, she forgives him, recognising that Jack wouldn't want his friend to keep blaming himself for what happened all those years ago.
In the Peter David novel Q-Squared, an alternate timeline is witnessed in which Jack Crusher is captain of the Enterprise after he took command of the Stargazer during its final mission and Picard was summarily demoted, although Jack and Beverly have divorced due to Wesley dying after falling out of a tree. In the course of the novel, Trelane reveals to this alternate Jack that he only survives in one timeline of all the timelines that exist in the multiverse order; in other words, he is alone, compared to the infinite copies of his friend, Jean-Luc Picard, and his ex, Beverly Howard. Through a series of events manipulated by Trelane, including the implication that Wesley died solely so that the universe could bring Picard and Beverly together- as well as Trelane 'tweaking' Jack's mind so that he believes Picard let him die in the 'main' universe on purpose rather than as an accident- this Jack Crusher goes insane and kills himself after killing his world's Beverly in front of the 'canon' Picard, destroying the only living version of Jack Crusher in all existence (this is debatable, however, since Trelane seems to have been purposefully tormenting Jack in order to further the chaos of merging timelines).