Where an ellipsis exists, is there a term for the missing text?

The elided material. I would be tempted to say the elision, but I haven't found evidence that the noun is actually used in that way --maybe we could pioneer that usage.

an omission or to omit a part of the sentence use an ellipsis

Ellipsis (plural ellipses; from the Ancient Greek: ἔλλειψις, élleipsis, "omission" or "falling short") is a series of dots that usually indicates an intentional omission of a word, sentence, or whole section from a text without altering its original meaning.

From the Wiki for Ellipsis

Since you're already familiar with orthographic term "ellipses" and (apparently) aren't looking for the editorial acts "elision", "excision", "truncation", "deletion" or @Malachi's perfectly suited "omission", maybe you're looking for a term which describes the elipsis' semantic role?

If so, I'd call ... the typographical analog to the "jump", "fold", or "spill line"; it teases or leads the reader into wanting to know more, so maybe you want to call it a "teaser".

The characters themselves are called ellipses (singular form ellipsis).

As for the term for the type of the characters... "truncation symbols"?