>Ok, that makes more sense. So, the family name (or surname or here in the U.S. lastname) would be 'Ivanov' and if someone had a lastname of Ivanova, they would know that she came from the family of Ivanov?

Yes.

(Or would the 'root' surname be Ivan?) I've seen many languages where words are gender specific (French, Spanish, et al), but not family names.

No. Surname Ivanov comes from name Ivan but it's irrelevant. In Russian language all nouns have gender.