Henry Archer was the primary developer of the first warp engine capable of reaching warp five, and was one of the principal designers of Enterprise NX-01. He was the father of Jonathan Archer and a good friend of Emory Erickson, the inventor of the transporter on Earth. (ENT: "Broken Bow", "Daedalus")
Prior to Cochrane's disappearance in 2119, Henry took his son Jonathan on a tour of the Warp Five Complex, and introduced the boy to Cochrane, Tasaki, and other scientists working at the facility. Jonathan later revealed he did not know the significance of his father's work at the time. (TOS: "Metamorphosis"; ENT: "Singularity")
In 2121, Henry helped his son build and fly a remote-controlled starship model. During this time, the elder Archer admonished his son's disrespect for the Vulcans, whom Jonathan believed were responsible for holding back Humanity's advancement into space. Although Henry insisted that Vulcan Ambassador Soval (whom Jonathan referred to as "Ambassador Pointy") had been very helpful, he also expressed confusion and disappointment at the Vulcans' refusal to share their technology, lamenting, "They have their reasons. God knows what they are."
Henry assisted Jonathan in flying his model starship on a beach, but the youth found keeping the craft in the air to be difficult. When the toy crashed, Henry advised his son that he "can't be afraid of the wind. Learn to trust it." The younger Archer took this advice to heart thirty years later, when he made the decision to abandon his prejudices against Vulcans and start trusting Sub-Commander T'Pol. (ENT: "Broken Bow")
Henry spent the last two years of his life experiencing frequent pain and hallucinations, talking to people who weren't there, and often not even recognizing his son or wife due to advanced Clarke's Disease. He died in 2124 before his warp five engine could be constructed, since the Vulcans had held back the engine's development program. As a result, Jonathan Archer's resentment towards Vulcans grew into a grudge which held until the 2150s. (ENT: "Broken Bow", "Cold Station 12")
In 2152, a biography on the life of Henry Archer was in the works. Jonathan Archer was asked to write a one-page preface for his father's biography, a task he put off for weeks as he found it difficult to sum up his father's life in one page. He ultimately completed it with feedback from T'Pol. One draft of the preface included the following:
"How does one measure a man's legacy? Is it defined [by] the works he's created....the technological advances that will forever alter the course of Human history? If so, then no man since Zefram Cochrane himself has made a more lasting contribution to the future of humankind than my father, Doctor Henry Archer. When I was about eight years old, my father took me on a tour of the Warp Five facility outside Bozeman, Montana. He introduced me to the people he worked with... scientists with names like Tasaki... and Cochrane. At the time, I didn't realize the significance of those names... or the significance of my father's work." (ENT: "Singularity")
As evidenced by the second draft of the script for ENT: "Broken Bow", this character was originally to have had the first name "William". The revised final draft script of "Broken Bow" simply described Henry Archer, when he appears in the teaser of that installment, as "in his 40's [...] wearing 22nd century civilian clothing." When he is thereafter shown with Zefram Cochrane, the script pointed out, "[He] is only a few years younger than when we saw him in the opening scene." 
It is unknown what became of Henry Archer's biography, though it was presumably published shortly after the events of "Singularity".
There appears to be some contradiction as to when Henry Archer died. Although "Cold Station 12" states that he passed away when Jonathan Archer was only twelve years old, dialogue in "Daedalus" suggests that he was still alive when the younger Archer entered flight training. In a scripted but ultimately omitted scene from an early script of ENT: "Home", Jonathan Archer visited his father's grave on Earth. According to Henry's grave marker, he was born in 2077 and died in 2133. (ENT Season 4 Blu-ray special features) This would have further contradicted Jonathan's claims that his father died when he was only twelve, in 2124.
Henry Archer also appears in the novelization of "Broken Bow".