Immortal children are a terrible
vampire tragedy plot device, which gives the Volturi a pretext on which to debate with "attack" the Cullens. They are unfortunate toddlers and/or small children who have been turned into sparklepires by particularly stupid and/or twisted vampires who think that child development is cute when it stops completely. In the real world such a complete lack of progress would call for emergency visits with the family doctor, and perhaps some occupational therapy, but in the vampire world they take the sensible path and just kill them off. It goes without saying that this concept is rather twisted, and unfortunately Stephenie Meyer has not been able to pull it off. (Chestburster Baby is a possible exception, but even she isn't done very well.) The only immortal child that is actually named is Vasilii.
It should be noted that this concept can be very effectively creepy - see Claudia in Interview With the Vampire. However attempts to make the concept cute or straightforwardly threatening are generally less successful. This is double true when the author is a talentless hack with no common sense.
Dangers Thereof Edit
Immortal children are apparently much stronger than adult vampires (perhaps the square/cube law coming into effect? Or most likely Meyer may have just been pulling this out of her ass.) as they somehow managed to devastate entire towns while whole covens of experienced adult vampires were supposedly guarding them . As a result of this devastation, the Volturi banned immortal children and killed them all, because there is no threat so serious as that of having your food source find out you exist. (This is a reminder: don't tell the cows about humans.)
And, honestly, the fact that Smeyer is cool with killing children in her books AND pedophillia just proves that she really might have some kind of mental issue, or that this is simply a failed attempt to add some creepy drama.
Immortal children were introduced abruptly (as in COMPLETELY OUT OF FUCKING NOWHERE) in Breaking Dawn, despite a total lack of any mention in the other books, to serve as a plot device, presumably because SMeyer's original plot device broke. (They're fragile. You have to keep your plot devices out of water and away from strong magnetic currents if you want them to last. Also try to handle them softly.) Their illegality allowed the Volturi to move against the Cullens, though the real goal was to capture them for Aro's menagerie. Unfortunately, proof that Impossibaby is not an immortal child arrives in time, and the epic battle that might have made this series even slightly worthwhile is averted. After all, heaven forbid this series have any real excitement or action in it - a black hole might appear out of the inherent contradiction and swallow the solar system.
It's worth pointing out that teenagers are not known for their maturity, but no punishment is meted out to vampires who turn them.