First, we should go over what being chosen is not. It is not a form of ethnocentrism.
I find it important to add this side note before continuing. The Chinese name for China is 中国, which literally means "Middle Kingdom." A sizable amount of Americans refer to America as the "greatest nation on Earth." Christians who believe in supersessionism opine that Judaism was replaced by Christianity, thereby making Christians the "new chosen people." If you have a culture, you will undoubtedly have ethnocentrism. It exists throughout the entirety of human history. Why are the Jews singled out with this accusation when every other group of people has been guilty of some form of ethnocentrism? I can guess the answer to the question, but it is a double standard regardless.
The notion of the Jewish people cannot be considered racist for the simple fact that Judaism is not a race. There are all sorts of Jews. There are American Jews, French Jews, Mexican Jews, Ethiopian Jews, Indian Jews, Japanese Jews, the list goes on and on. With such a racial diversity, Jews cannot be classified as a race. Furthermore, the Jewish people are non-exclusionary in the sense that anybody can convert to Judaism.
Being part of the "chosen people" does not even mean that Jews are better than non-Jews. Just like everybody else, Jews have made mistakes. The Hebrew Bible is a constant reminder of that fact. As soon as the Jews left Egypt, there was the Golden Calf incident. Jews have been exiled twice from Israel. Even on the individual level, Jews are imperfect. We're human, just like everybody else, which is why Judaism has a strong belief in the dignity of man.
So if being chosen does not have anything to do with racism or superiority, what does it mean? For what reason were the Jews chosen?
G-d chose the Jews as a spiritual vocation, to be "a holy nation (Exodus 19:6)," as it were. What that means is that Jews were not chosen for privileges; they were chosen to undertake more responsibilities. Non-Jews are not required to observe the Sabbath, the dietary laws (kashrut), or make sure that your wool and linen is not mixed. The Torah and mitzvoth are responsibilities solely given to the Jews. In addition to these obligations, Jews have obligations to humankind by changing the world for the better. Jews are meant to be a "light unto nations (Isaiah 42:6)." In this respect, Jews are to be a role model for the rest of the world in order that together, we can actualize Messianic prophecy and create a more just, compassionate society.
This post-Shabbat blog was inspired by a sermon I heard on August 26, 2011.