In Chapter Seven of Tuck Everlasting, the Tuck family explains to Winnie how they became immortal. Having stopped in the woods surrounding Treegap to rest, they all drank from a spring... including their horse. As years passed, various bizarre incidences started to indicate that something was not quite right with the family. Jesse was sawing large branches off of a tree when he lost his balance and fell out of it, landing on his head but experiencing no injury. The next strange event involved the horse. Hunters who were passing by the Tuck residence shot the horse at sunset, believing it to be a deer, but "The bullet went right on through him, and didn't hardly even leave a mark." After several more near-catastrophic injuries that didn't affect the Tucks, they realized they were going to live forever.
(On a side note, it is also worth mentioning that The Man in the Yellow Suit steals the horse much later in the book and rides it to the Fosters' home, although it is eventually returned to the Tucks to continue to live out its eternal life.)