Henry Archer was the primary developer of the first warp engine capable of reaching warp five, and 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 would take this advise to heart 30 years later when he made the decision to abandon his prejudices against Vulcans and start trusting Subcommander 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")
Henry Archer was played by actor Mark Moses in his second of two Star Trek appearances. The shirt worn by Moses in the final scene of "Broken Bow" was sold off on the It's A Wrap! sale and auction on eBay. 
A draft version of the script for "Broken Bow" simply describes Henry Archer as "in his 40's [...] wearing 22nd century civilian clothing." 
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 he died. Although "Cold Station 12" stated he passed away when Jonathan was only 12 years-old, dialog in "Daedalus" suggests that he was still alive when the younger Archer entered flight training. In a deleted scene from 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. 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".