How children born abroad to a US citizen mother or father acquire or derive US citizenship through their parents, their mother or father. How children used to derive US citizenship under the old laws (which is still relevant because it applies to those children who were born during those timeframes).
Please see charts on acquisition/derivation of citizenship here: http://www.ilrc.org/resources/naturalization-quick-reference-charts
Chart A:
https://dl.dropboxusercontent.com/u/64881744/1.CHILD_CITIZENSHIP_CHART_1_natz_chart-a-2014-11-12.pdf
Chart B:
https://dl.dropboxusercontent.com/u/64881744/1.CHILD_CITIZENSHIP_CHART_2_OUT_OF_WEDLOCK_natz_chart-b-2014-11.12.pdf
Chart C: https://dl.dropboxusercontent.com/u/64881744/1.CHILD_CITIZENSHIP_CHART_3_natz_chart-c-2015_3-4.ak_.pdf