Elizabeth was a binary clone created from the DNA of T'Pol, a Vulcan female, and Charles Tucker, a Human male, both of whom served aboard the starship Enterprise NX-01. She was created by the xenophobic extremist group Terra Prime, with the intent of using her as a rallying point for their anti-alien views. She was named after Elizabeth Tucker, Commander Tucker's sister who died in the Xindi attack in 2153.
Elizabeth's existence was first discovered in January of 2155, when former Terra Prime member Susan Khouri gave her life to deliver a vial of Elizabeth's hair to T'Pol. Running a genetic analysis, Doctor Phlox discovered that the hair belonged to an infant female who was the offspring of Tucker and T'Pol. John Frederick Paxton, Terra Prime's leader, used Elizabeth as an example of the genetic pollution that could occur if Humans and aliens continued to coexist. He then threatened to destroy Starfleet Command with the verteron array on Mars unless every alien left the solar system. Ultimately, Paxton's plot was foiled by the crew of Enterprise NX-01, and Elizabeth was rescued. Sadly, Elizabeth died soon afterward, due to a fever and an elevated white blood cell count. Elizabeth's importance as the first known alien-Human hybrid was such that, following her death, the delegates from the conference for the Coalition of Planets asked to attend the child's funeral services.
Although it was initially believed that the infant died due to the fundamental incompatibility of Human and Vulcan genes, Phlox later discovered that Elizabeth had died due to a flaw in the cloning procedure used to create her, and that there was no reason why a Human and a Vulcan could not have a child. (ENT: "Demons", "Terra Prime")
This character was invented to live or die based on whether a fifth season of Star Trek: Enterprise was given the go-ahead or not. While the episode "Terra Prime" was in development, Manny Coto had the idea that, if another season was indeed approved, the baby would live, but if the series was canceled, the child would die. The writers hoped the baby would live, as they expected it would make for "a great storyline." However, the death of Baby Elizabeth echoes the fact that Enterprise was canceled at the end of its fourth season. ("Before Her Time: Decommissioning Enterprise, Part Three: Final Approach", ENT Season 4 Blu-ray special features)