According to halakha

According to halakha, the oldest normative definition used by Jews for self-identification, a person is matrilineally a Jew by birth, or becomes one through conversion to Judaism.
...Reform and Liberal Judaism do not accept the halakhic rules as binding, and accept a child of one Jewish parent, whether father or mother, as Jewish if the parents raise the child as a Jew and the child fosters a Jewish identity...
http://en.wikipedia.org/wiki/Who_is_a_Jew%3F

I wasn't surprised to find some websites and blogs write articles about the Jewish lawmaker, Gabrielle Giffords. After all, The Reform movement accepts her as being Jewish because of her Jewish father. But, I was surprised to find some websites and blogs leaning to an Orthodox point of view refer to her as Jewish. Perhaps Aish.com got it right when she was described as follows.

Ms. Giffords narrowly won, and became the first Jewishly identified representative from her state.
Rep. Giffords did not always consider herself Jewish. With a Christian mother and a Jewish father, Rep. Giffords began to identify herself as Jewish after a political visit to Israel ten years ago.
http://www.aish.com/ci/s/The_Shooting_of_Gabrielle_Giffords.html

And here is a part of a comment posted at the end of another article.

Her mother is not Jewish.
She did not CONVERT according to Jewish Law.
HENCE, she is NOT Jewish.
She may be a RIGHTEOUS GENTILE, but certainly NOT Jewish...
Unless we let everything and everyone decide who is Jewish or not accept Jewish Law and Tradition of 3,000 years.
It would be like letting a microbiologist decide how to better perform orthopedic surgery, a mechanic decide how to run a biology experiment, a painter deciding how to graft a plant rather than a botanist.

May she and the other victims have speedy recoveries.