(written from a Production point of view)
Jack Archer was the great-grandfather of Jonathan Archer. He never appears nor is referred to in canon, but he was referenced in ultimately deleted dialogue from the final draft script of the Star Trek: Enterprise episode "Twilight".
In that script, Jonathan Archer told T'Pol about his great-grandfather Jack, for the first time, in 2155 of an alternate timeline, stating that Jack had had several children (obviously including Jonathan's grandfather). Jack had developed a memory-impairing degenerative neural disorder for which there had been no cure until after he had died. According to stories Jonathan had heard, Jack repeatedly woke up in the morning and wandered the family house, searching for his wife, whom Jack's children had to remind him had been dead for twenty years. With a touch of irony, Jonathan Archer implied a connection between himself, whom T'Pol had essentially told to "stay out of the way", and Jack Archer, about whom Jonathan concluded, "I wouldn't be surprised if Jack's only responsibility was to 'stay out of the way.'"
"Jack" was originally to have been a colloquial form of addressing the character who eventually became Jonathan Archer, as he was originally to have been named "Jackson Archer".