Americans have a strong belief in "rights". The Constitution of the U.S., which is the document that contains the country's most important laws, gives Americans several rights such as the right to carry weapons and the right to free speech.
So what does it mean to "have the right to" do something? It means that you are allowed to do it, and no one should be allowed to make you stop doing it. So the sentence:
Shouldn't a woman have the right to choose what she does with her own life?
...means that women should be allowed to choose whether to stay home or go to work.
Other "rights" that you may hear people talk about include:
- basic human rights (things like food, clean water, and not being made into a slave by anyone)
- civil rights (not being discriminated against or treated badly by the government)
- the right to remain silent (for someone who is arrested)